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Journal articles on the topic 'Graph X'

Author: Grafiati

Published: 5 June 2025

Last updated: 25 June 2025

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1

Firmansah, Fery. "The Odd Harmonious Labeling of Layered Graphs." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 2 (2023): 339. http://dx.doi.org/10.31764/jtam.v7i2.12506.

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Graphs that have the properties of odd harmonious labeling are odd harmonious graphs. The research objective of this paper is to obtain odd harmonious labeling on layered graph C(x,y) and layered graph D(x,y). The research used in this paper is a qualitative method. The research flow consists of data collection, processing, and analysis. The data collection stage consists of constructing the definition of the new class graph, the data processing stage consists of constructing the vertex labeling and edge labeling, and the data analysis stage consists of constructing the theorem and proving it. The research results show that the layered graph C(x,y) and layered graph D(x,y) fulfill odd harmonious labeling. Such that the layered graph C(x,y) and layered graph D(x,y) are odd harmonious graphs. The benefit of this research is to add new properties of odd harmonious graphs. In addition, it does not rule out the possibility that this research can be developed again both in theory and application.

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ATMINAS, AISTIS, VADIM V. LOZIN, SERGEY KITAEV, and ALEXANDR VALYUZHENICH. "UNIVERSAL GRAPHS AND UNIVERSAL PERMUTATIONS." Discrete Mathematics, Algorithms and Applications 05, no. 04 (2013): 1350038. http://dx.doi.org/10.1142/s1793830913500389.

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Let X be a family of graphs and Xn the set of n-vertex graphs in X. A graph U(n) containing all graphs from Xn as induced subgraphs is called n-universal for X. Moreover, we say that U(n) is a propern-universal graph for X if it belongs to X. In the present paper, we construct a proper n-universal graph for the class of split permutation graphs. Our solution includes two ingredients: a proper universal 321-avoiding permutation and a bijection between 321-avoiding permutations and symmetric split permutation graphs. The n-universal split permutation graph constructed in this paper has 4n3 vertices, which means that this construction is order-optimal.

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3

Guo, Jin, Tongsuo Wu, and Meng Ye. "Complemented graphs and blow-ups of Boolean graphs, with applications to co-maximal ideal graphs." Filomat 29, no. 4 (2015): 897–908. http://dx.doi.org/10.2298/fil1504897g.

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For a set X, let 2X be the power set of X. Let BX be the Boolean graph, which is defined on the vertex set 2X \ {X, ?}, with M adjacent to N if M ? N = ?. In this paper, several purely graph-theoretic characterizations are provided for blow-ups of a finite or an infinite Boolean graph (respectively, a preatomic graph). Then the characterizations are used to study co-maximal ideal graphs that are blow-ups of Boolean graphs (pre-atomic graphs, respectively).

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4

Zupan, J. "GRAPH X." Journal of Chemical Information and Modeling 28, no. 2 (1988): 118–19. http://dx.doi.org/10.1021/ci00058a013.

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5

Liu, Shunyi. "Generalized Permanental Polynomials of Graphs." Symmetry 11, no. 2 (2019): 242. http://dx.doi.org/10.3390/sym11020242.

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The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let G be a graph with adjacency matrix A ( G ) and degree matrix D ( G ) . The generalized permanental polynomial of G is defined by P G ( x , μ ) = per ( x I − ( A ( G ) − μ D ( G ) ) ) . In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efficient for distinguishing graphs. Furthermore, we can write P G ( x , μ ) in the coefficient form ∑ i = 0 n c μ i ( G ) x n − i and obtain the combinatorial expressions for the first five coefficients c μ i ( G ) ( i = 0 , 1 , ⋯ , 4 ) of P G ( x , μ ) .

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6

Amreen, J., and S. Naduvath. "Non-inverse signed graph of a group." Carpathian Mathematical Publications 16, no. 2 (2024): 565–74. https://doi.org/10.15330/cmp.16.2.565-574.

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Let $G$ be a group with binary operation $\ast$. The non-inverse graph (in short, $i^*$-graph) of $G$, denoted by $\Gamma$, is a simple graph with vertex set consisting of elements of $G$ and two vertices $x, y \in \Gamma$ are adjacent if $x$ and $y$ are not inverses of each other. That is, $x- y$ if and only if $x\ast y \neq i_G \neq y \ast x$, where $i_G$ is the identity element of $G$. In this paper, we extend the study of $i^\ast$-graphs to signed graphs by defining $i^\ast$-signed graphs. We characterize the graphs for which the $i^\ast$-signed graphs and negated $i^\ast$-signed graphs are balanced, sign-compatible, consistent and $k$-clusterable. We also obtain the frustration index of the $i^\ast$-signed graph. Further, we characterize the homogeneous non-inverse signed graphs and study the properties like net-regularity and switching equivalence.

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S., Vimala, and Priyanka K. "Topologized Hamiltonian and Complete Graph." Asian Research Journal of Mathematics 4, no. 2 (2017): 1–10. https://doi.org/10.9734/ARJOM/2017/33109.

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Topological graph theory deals with embedding the graphs in Surfaces, and the graphs considered as a topological spaces. The concept topology extended to the topologized graph by the S<sub>1 </sub>space and the boundary of every vertex and edge. The space is S<sub>1 </sub>if every singleton in the topological space is either open or closed. Let G be a graph with n vertices and e edges and a topology defined on graph is called topologized graph if it satisfies the following: Every singleton is open or closed and For all x X, | ∂(x) |≤ 2, where ∂(x) is the boundary of a point x. This paper examines some results about the topological approach of the Complete Graph, Path, Circuit, Hamiltonian circuit and Hamiltonian path. And the results were generalized through this work.

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8

Pushpam, P. Roushini Leely, and D. Yokesh. "Differentials in certain classes of graphs." Tamkang Journal of Mathematics 41, no. 2 (2010): 129–38. http://dx.doi.org/10.5556/j.tkjm.41.2010.664.

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Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

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9

Sutardji, Nurma Ariska, Liliek Susilowati, and Utami Dyah Purwati. "Dimensi Metrik Kuat Lokal Graf Hasil Operasi Kali Kartesian." Contemporary Mathematics and Applications (ConMathA) 1, no. 2 (2020): 64. http://dx.doi.org/10.20473/conmatha.v1i2.17383.

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The strong local metric dimension is the development result of a strong metric dimension study, one of the study topics in graph theory. Some of graphs that have been discovered about strong local metric dimension are path graph, star graph, complete graph, cycle graphs, and the result corona product graph. In the previous study have been built about strong local metric dimensions of corona product graph. The purpose of this research is to determine the strong local metric dimension of cartesian product graph between any connected graph G and H, denoted by dimsl (G x H). In this research, local metric dimension of G x H is influenced by local strong metric dimension of graph G and local strong metric dimension of graph H. Graph G and graph H has at least two order.

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10

Rana, A. "On the Total Vertex Irregular Labeling of Proper Interval Graphs." Journal of Scientific Research 12, no. 4 (2020): 537–43. http://dx.doi.org/10.3329/jsr.v12i4.45923.

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A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V, E) with vertex set V and edge set E, a labeling Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is deﬁned to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

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11

ZHOU, JIN-XIN, and YAN-QUAN FENG. "TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER." Journal of the Australian Mathematical Society 88, no. 2 (2010): 277–88. http://dx.doi.org/10.1017/s1446788710000066.

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AbstractA graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

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12

Aniyan, Achu, and Sudev Naduvath. "Subspace graph topological space of graphs." Proyecciones (Antofagasta) 42, no. 2 (2023): 521–32. http://dx.doi.org/10.22199/issn.0717-6279-5386.

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A graph topology defined on a graph G is a collection 𝒯 of subgraphs of G which satisfies the properties such as K0, G ∈ 𝒯 and 𝒯 is closed under arbitrary union and finite intersection. Let (X, T) be a topological space and Y ⊆ X then, TY = {U ∩ Y : U ∈ T} is a topological space called a subspace topology or relative topology defined by T on Y. In this P1 we discusses the subspace or the relative graph topology defined by the graph topology 𝒯 on a subgraph H of G. We also study the properties of subspace graph topologies, open graphs, d-closed graphs and nbd-closed graphs of subspace graph topologies.

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13

Simanjuntak, Rinovia, and Aholiab Tritama. "Distance Antimagic Product Graphs." Symmetry 14, no. 7 (2022): 1411. http://dx.doi.org/10.3390/sym14071411.

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A distance antimagic graph is a graph G admitting a bijection f:V(G)→{1,2,…,|V(G)|} such that for two distinct vertices x and y, ω(x)≠ω(y), where ω(x)=∑y∈N(x)f(y), for N(x) the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs.

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14

Muhammad Imran. "Computation of Lucky Number of Comb Graphs Cfw, Cgw, Chw and Triangular Snake, Alternate Triangular Snake Graphs." Power System Technology 48, no. 1 (2024): 1381–94. https://doi.org/10.52783/pst.399.

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Let λ:V(Ω)→{1, 2,..., z} be a mapping of vertices of a graph Ω. Let S(x) denote the sum of labels of the neighbors of the vertex x in Ω. If vertex x has degree zero, we put S(x)=0. A mapping λ is categorized as lucky labeling if S(x)=S(y) for every pair of adjacent vertices x and y. The lucky number of graph Ω, denoted by η (Ω), is the least positive integer z used to label vertices to form lucky labeling. In this paper, we demonstrate that different families of comb graph and snake graph are lucky labeled graphs. We also calculate the exact value of the lucky number for the aforementioned graphs.

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15

Tacbobo, Teresa L. "The Generator Graph of a Group." European Journal of Pure and Applied Mathematics 16, no. 3 (2023): 1894–901. http://dx.doi.org/10.29020/nybg.ejpam.v16i3.4863.

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This paper presents a way to represent a group using a graph, which involves the concept of a generator element of a group. The graph representing a group is called the generator graph. In the generator graph, the vertices correspond to the elements of the group, and two vertices, x and y, are connected by an edge if either x or y serves as a generator for the group. The paper investigates some properties of these generator graphs and obtains the generator graphs for specific groups. Additionally, it explores the relationship between the generator graph of a group and the generating graph introduced by Lucchini et al. in their work [7].

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16

Darmaji and N. Azahra. "The mixed metric dimension of wheel-like graphs." Journal of Physics: Conference Series 2157, no. 1 (2022): 012010. http://dx.doi.org/10.1088/1742-6596/2157/1/012010.

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Abstract Consider the graph G = (V, E). It is a connected graph. It is a simple graph too. A node w ∈ V, then we call vertex, determined two elements of graph. There are vertices and edges of graphs. Any two vertices x, y ∈ E ∪ V if d(w, x) ≠ d(w, y), which d(w, x) and d(w, y) is the mixed distance of the element w (vertices or edges) in graph G. A set of vertices in a graph G is represented by the symbol Rm that defines a mixed metric generator for G, if the elements of vertices or edges are stipulated by several vertex set of Rm . There’s a chance that some mixed metric generators have varied cardinality. We choose one whose the minimum cardinality and it is called the mixed metric dimension of graph G, denoted by dimm (G). This research examines the mixed metric dimension of gear Gn , helm Hn , sunflower SFn , and friendship graph Frn . We call these graphs by wheel-like graphs. Our findings include the mixed metric dimension of gear graph Gn of order n ≥ 4 is dimm (Gn ) = n, helm graph Hn of order n ≥ 4 is dimm (Hn ) = n, sunflower graph SFn of order n ≥ 5 is dimm (SFn ) = n and friendship graph Frn of order n ≥ 3 is dimm (Frn ) = 2n.

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17

Seifter, Norbert, and Wolfgang Woess. "Approximating graphs with polynomial growth." Glasgow Mathematical Journal 42, no. 1 (2000): 1–8. http://dx.doi.org/10.1017/s001708950001003x.

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Let X be an infinite, locally finite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an infinite sequence of finite graphs satisfying growth conditions which are closely related to growth properties of the infinite graph X.1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

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Simanjuntak, Rinovia, Tamaro Nadeak, Fuad Yasin, Kristiana Wijaya, Nurdin Hinding, and Kiki Ariyanti Sugeng. "Another Antimagic Conjecture." Symmetry 13, no. 11 (2021): 2071. http://dx.doi.org/10.3390/sym13112071.

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An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

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Sy, Syafrizal, Rinovia Simanjuntak, Tamaro Nadeak, Kiki Ariyanti Sugeng, and Tulus Tulus. "Distance antimagic labeling of circulant graphs." AIMS Mathematics 9, no. 8 (2024): 21177–88. http://dx.doi.org/10.3934/math.20241028.

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A distance antimagic labeling of graph $ G = (V, E) $ of order $ n $ is a bijection $ f:V(G)\rightarrow \{1, 2, \ldots, n\} $ with the property that any two distinct vertices $ x $ and $ y $ satisfy $ \omega(x)\ne\omega(y) $, where $ \omega(x) $ denotes the open neighborhood sum $ \sum_{a\in N(x)}f(a) $ of a vertex $ x $. In 2013, Kamatchi and Arumugam conjectured that a graph admits a distance antimagic labeling if and only if it contains no two vertices with the same open neighborhood. A circulant graph $ C(n; S) $ is a Cayley graph with order $ n $ and generating set $ S $, whose adjacency matrix is circulant. This paper provides partial evidence for the conjecture above by presenting distance antimagic labeling for some circulant graphs. In particular, we completely characterized distance antimagic circulant graphs with one generator and distance antimagic circulant graphs $ C(n; \{1, k\}) $ with odd $ n $.

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20

Lo Faro, Giovanni, Salvatore Milici, and Antoinette Tripodi. "Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n." Mathematics 8, no. 10 (2020): 1755. http://dx.doi.org/10.3390/math8101755.

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If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.

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Li, Ruijuan, Xiaoting An, and Xinhong Zhang. "The (1, 2)-step competition graph of a hypertournament." Open Mathematics 19, no. 1 (2021): 483–91. http://dx.doi.org/10.1515/math-2021-0047.

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Abstract In 2011, Factor and Merz [Discrete Appl. Math. 159 (2011), 100–103] defined the ( 1 , 2 ) \left(1,2) -step competition graph of a digraph. Given a digraph D = ( V , A ) D=\left(V,A) , the ( 1 , 2 ) \left(1,2) -step competition graph of D, denoted C 1 , 2 ( D ) {C}_{1,2}\left(D) , is a graph on V ( D ) V\left(D) , where x y ∈ E ( C 1 , 2 ( D ) ) xy\in E\left({C}_{1,2}\left(D)) if and only if there exists a vertex z ≠ x , y z\ne x,y such that either d D − y ( x , z ) = 1 {d}_{D-y}\left(x,z)=1 and d D − x ( y , z ) ≤ 2 {d}_{D-x}(y,z)\le 2 or d D − x ( y , z ) = 1 {d}_{D-x}(y,z)=1 and d D − y ( x , z ) ≤ 2 {d}_{D-y}\left(x,z)\le 2 . They also characterized the (1, 2)-step competition graphs of tournaments and extended some results to the ( i , j ) \left(i,j) -step competition graphs of tournaments. In this paper, the definition of the (1, 2)-step competition graph of a digraph is generalized to a hypertournament and the (1, 2)-step competition graph of a k-hypertournament is characterized. Also, the results are extended to ( i , j ) \left(i,j) -step competition graphs of k-hypertournaments.

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Hossein-Zadeh, Samaneh, Ali Iranmanesh, Mohammad Ali Hosseinzadeh, and Mark L. Lewis. "On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups." Canadian Mathematical Bulletin 58, no. 1 (2015): 105–9. http://dx.doi.org/10.4153/cmb-2014-058-8.

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Abstract.The prime vertex graph, Δ(X), and the common divisor graph, Γ(X), are two graphs that have been deûned on a set of positive integers X. Some properties of these graphs have been studied in the cases where either X is the set of character degrees of a group or X is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

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B. Basavanagou and Shruti Policepatil. "INTEGRITY OF SOME DERIVED GRAPHS." Jnanabha 51, no. 01 (2021): 182–200. http://dx.doi.org/10.58250/jnanabha.2021.51123.

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One of the best parameter to measure the stability of a network is integrity as it takes into account both the amount of work done to damage the network and how badly the network is damaged. The integrity I(G) of a graph G is a measure of network vulnerability and is defined by I(G) = min{|S | + m(G − S )}, where S and m(G − S ) denote the subset of V and order of the largest component of G − S , respectively. In this paper, we study the integrity of line graph, jump graph, para-line graph of some standard graph families. In this way, we establish the relationship between integrity of basic graphs and integrity of their derived graphs. Also, we characterize few graphs having equal integrity values as that of derived graphs of same structured graphs. Further, we determine the integrity of generalized xyz-point-line transformation graphs Tˣʸ⁺ (G), Tˣʸ⁻(G), where x, y ∈ {0, 1, +, −} and Tˣʸ¹ (G), where x, y ∈ {0, 1, +, −}

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Ali, Nasir, Hafiz Muhammad Afzal Siddiqui, Muhammad Imran Qureshi, et al. "Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs." Symmetry 16, no. 7 (2024): 930. http://dx.doi.org/10.3390/sym16070930.

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This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z(RE)\{[0]}=RE\{[0],[1]}, where RE={[x] :x∈R} and [x]={y∈R : ann(x)=ann(y)} is called a compressed zero-divisor graph, denoted by ΓER. An edge is formed between two vertices [x] and [y] of Z(RE) if and only if [x][y]=[xy]=[0], that is, iff xy=0. For a ring R, graph G is said to be realizable as ΓER if G is isomorphic to ΓER. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance.

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Feng, Yan-Quan, and Jin Ho Kwak. "Cubic symmetric graphs of order twice an odd prime-power." Journal of the Australian Mathematical Society 81, no. 2 (2006): 153–64. http://dx.doi.org/10.1017/s1446788700015792.

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AbstractAn automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

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Yanamandram, Venkatakrishnan Balasubramanian, C. Natarajan, and S. K. Ayyaswamy. "X− Dominating colour transversals in graphs." Boletim da Sociedade Paranaense de Matemática 34, no. 2 (2015): 99–105. http://dx.doi.org/10.5269/bspm.v34i2.22997.

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Let G = (X, Y,E) be a bipartite graph. A X-dominating set D ⊆X is called a X−dominating colour transversal set of a graph G if D isa transversal of at least one $chi$−partition of G.The minimum cardinal-ity of a X−dominating colour transversal set is called X−dominatingcolour transversal number and is denoted by $chi_{dct}(G)$. We find thebounds of X−dominating colour transversal number and characterizethe graphs attaining the bound.

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Alrowaili, Dalal Awadh, Uzma Ahmad, Saira Hameeed, and Muhammad Javaid. "Graphs with mixed metric dimension three and related algorithms." AIMS Mathematics 8, no. 7 (2023): 16708–23. http://dx.doi.org/10.3934/math.2023854.

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<abstract><p>Let $ G = (V, E) $ be a simple connected graph. A vertex $ x\in V(G) $ resolves the elements $ u, v\in E(G)\cup V(G) $ if $ d_G(x, u)\neq d_G(x, v) $. A subset $ S\subseteq V(G) $ is a mixed metric resolving set for $ G $ if every two elements of $ G $ are resolved by some vertex of $ S $. A set of smallest cardinality of mixed metric generator for $ G $ is called the mixed metric dimension. In this paper trees and unicyclic graphs having mixed dimension three are classified. The main aim is to investigate the structure of a simple connected graph having mixed dimension three with respect to the order of graph, maximum degree of basis elements and distance partite sets of basis elements. In particular to find necessary and sufficient conditions for a graph to have mixed metric dimension 3. Moreover three separate algorithms are developed for trees, unicyclic graphs and in general for simple connected graph $ J_{n}\ncong P_{n} $ with $ n\geq 3 $ to determine "whether these graphs have mixed dimension three or not?". If these graphs have mixed dimension three, then these algorithms provide a mixed basis of an input graph.</p></abstract>

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Tapeing, Aziz B., and Ladznar S. Laja. "CO-SEGREGATED POLYNOMIAL OF GRAPHS." Advances and Applications in Discrete Mathematics 40, no. 1 (2023): 101–12. http://dx.doi.org/10.17654/0974165823059.

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A graph $G$ is co-segregated if $\text{deg}_G(x)=\text{deg}_G(y),$ then $xy \in E(G)$. The co-segregated polynomial of a graph $G$ of order $n$ is given by $CoS(G,x)=\sum_{k=1}^{n}C(k)x^k$, where $C(k)$ is the number of co-segregated subgraphs of $G$ of order $k$. We characterize a co-segregated subgraph of a graph and also of a graph under some binary operations. Using these characterizations, we obtain co-segregated polynomials of such graphs.

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Aniyan, Achu, and Sudev Naduvath. "Subspace spanning graph topological spaces of graphs." Proyecciones (Antofagasta) 42, no. 2 (2023): 479–88. http://dx.doi.org/10.22199/issn.0717-6279-5674.

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A collection of spanning subgraphs TS, of a graph G is said to be a spanning graph topology if it satisfies the three axioms: Nn, K0 ∈ TS where, n = |V (G)|, the collection is closed under any union and finite intersection. Let (X, T) be a topological space in point set topology and Y ⊆ X then, TY = {U ∩ Y : U ∈ T} is a topological space called a subspace topology or relative topology defined by T on Y . In this paper we discusses the subspace spanning graph topology defined by the graph topology TS on a spanning subgraph H of G.

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30

Rochanakul, Penying, Hatairat Yingtaweesittikul, and Sayan Panma. "Formulas for the Number of Weak Homomorphisms from Paths to Rectangular Grid Graphs." Symmetry 17, no. 4 (2025): 497. https://doi.org/10.3390/sym17040497.

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A weak homomorphism from graph G to graph H is a mapping f:V(G)→V(H), where either f(x)=f(y) or {f(x),f(y)}∈E(H) hold for all {x,y}∈E(G). A rectangular grid graph is formed by taking the Cartesian product of two paths. Counting weak homomorphisms is a fundamental problem in graph theory. In this paper, we present formulas for calculating the number of weak homomorphisms from paths to rectangular grid graphs. This count directly corresponds to the number of partial walks with length m within the rectangular grid graph, offering a combinatorial solution to this enumeration problem.

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Hao, Shangjing, Guo Zhong, and Xuanlong Ma. "Notes on the Co-prime Order Graph of a Group." Proceedings of the Bulgarian Academy of Sciences 75, no. 3 (2022): 340–48. http://dx.doi.org/10.7546/crabs.2022.03.03.

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The co-prime order graph of a group $$G$$ is the graph with vertex set $$G$$, and two distinct elements $$x,y\in G$$ are adjacent if gcd$$(o(x),o(y))$$ is either $$1$$ or a prime, where $$o(x)$$ and $$o(y)$$ are the orders of $$x$$ and $$y$$, respectively. In this paper, we characterize finite groups whose co-prime order graphs are complete and classify finite groups whose co-prime order graphs are planar, which generalizes some results by Banerjee [3]. We also compute the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group and a generalized quaternion group, which answers a question by Banerjee [3].

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32

Septory, Brian Juned, Liliek Susilowaty, Dafik, V. Lokehsa, and G. Nagamani. "On the study of Rainbow Antimagic Connection Number of Corona Product of Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 271–85. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4520.

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Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each vertex under the label function f : V → {1, 2, . . . , |V (G)|} such that the associated weight of an edge uv ∈ E(G), namely w(xy) = f(x) + f(y), has distinct weight. A path P in the vertex-labeled graph G is said to be a rainbow path if for every two edges xy, x′y ′ ∈ E(P) satisfies w(xy) ̸= w(x ′y ′ ). The function f is called a rainbow antimagic labeling of G if for every two vertices x and y of G, there exists a rainbow x − y path. When we assign each edge xy with the color of the edge weight w(xy), thus we say the graph G admits a rainbow antimagic coloring. The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors induced from all edge weight of antimagic labeling. In this paper, we will study the rac(G) of the corona product of graphs. By the corona product of graphs G and H, denoted by G ⊙ H, we mean a graph obtained by taking a copy of graph G and n copies of graph H, namely H1, H2, ..., Hn, then connecting vertex vi from the copy of graph G to every vertex on graph Hi , i = 1, 2, 3, . . . , n. In this paper, we show the exact value of the rainbow antimagic connection number of Tn ⊙ Sm where Tn ∈ {Pn, Sn, Sn,p, Fn,3}.

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Ma, Xuesong, and Ruji Wang. "Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150." Algebra Colloquium 15, no. 03 (2008): 379–90. http://dx.doi.org/10.1142/s1005386708000370.

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Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

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Hamzeh, Asma, and Ali Ashrafi. "Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups." Filomat 31, no. 16 (2017): 5323–34. http://dx.doi.org/10.2298/fil1716323h.

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Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

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Gein, Pavel A. "ON CHROMATIC UNIQUENESS OF SOME COMPLETE TRIPARTITE GRAPHS." Ural Mathematical Journal 7, no. 1 (2021): 38. http://dx.doi.org/10.15826/umj.2021.1.004.

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Let \(P(G, x)\) be a chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are called chromatically equivalent iff \(P(G, x) = H(G, x)\). A graph \(G\) is called chromatically unique if \(G\simeq H\) for every \(H\) chromatically equivalent to \(G\). In this paper, the chromatic uniqueness of complete tripartite graphs \(K(n_1, n_2, n_3)\) is proved for \(n_1 \geqslant n_2 \geqslant n_3 \geqslant 2\) and \(n_1 - n_3 \leqslant 5\).

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36

BARATI, Z., K. KHASHYARMANESH, F. MOHAMMADI, and KH NAFAR. "ON THE ASSOCIATED GRAPHS TO A COMMUTATIVE RING." Journal of Algebra and Its Applications 11, no. 02 (2012): 1250037. http://dx.doi.org/10.1142/s0219498811005610.

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Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by ΓS(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of ΓS(R). Moreover, we will improve and generalize some results for the total and the unit graphs.

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KHEIRABADI, M., and A. R. MOGHADDAMFAR. "RECOGNIZING SOME FINITE SIMPLE GROUPS BY NONCOMMUTING GRAPH." Journal of Algebra and Its Applications 11, no. 04 (2012): 1250077. http://dx.doi.org/10.1142/s0219498812500776.

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Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.

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38

Basher, M., and Muhammad Kamran Siddiqui. "On even-odd meanness of super subdivision of some graphs." Proyecciones (Antofagasta) 41, no. 5 (2022): 1213–28. http://dx.doi.org/10.22199/issn.0717-6279-5302.

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Graph Labeling is a significant area of graph theory that is used in a variety of applications like coding hypothesis, x-beam crystallography, radar, cosmology, circuit design, correspondence network tending to, and database administration. This study provides a general overview of graph naming in heterogeneous fields, however it primarily focuses on graph subdivision. The even vertex odd meanness of super subdivide of various graphs is discussed in this study. The graphs generated by super subdivided of path, cycle, comb, crown, and planar grid are even-odd mean graphs, according to our proof.

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39

Britton, Tom, and Pieter Trapman. "Maximizing the Size of the Giant." Journal of Applied Probability 49, no. 04 (2012): 1156–65. http://dx.doi.org/10.1017/s0021900200012948.

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Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring ‘on’ the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X i with E(X i )=µ, and where there is an edge between i and j with probability 1-e-X i X j /(µ n), independently of other edges. The second model is the thinned configuration model in which the n vertices of the ground graph have i.i.d. ground degrees, distributed as D, with E(D) = µ. The graph of interest is obtained by deleting edges independently with probability 1-p. In both models the fraction of vertices in the largest connected component converges in probability to a constant 1-q, where q depends on X or D and p. We investigate for which distributions X and D with given µ and p, 1-q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

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40

Dukes, Peter, and Alan C. H. Ling. "Asymptotic Existence of Resolvable Graph Designs." Canadian Mathematical Bulletin 50, no. 4 (2007): 504–18. http://dx.doi.org/10.4153/cmb-2007-050-x.

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AbstractLet v ≥ k ≥ 1 and λ ≥ 0 be integers. A block design BD(v, k, λ) is a collection of k-subsets of a v-set X in which every unordered pair of elements from X is contained in exactly λ elements of . More generally, for a fixed simple graph G, a graph design GD(v, G, λ) is a collection of graphs isomorphic to G with vertices in X such that every unordered pair of elements from X is an edge of exactly λ elements of . A famous result of Wilson says that for a fixed G and λ, there exists a GD(v, G, λ) for all sufficiently large v satisfying certain necessary conditions. A block (graph) design as above is resolvable if can be partitioned into partitions of (graphs whose vertex sets partition) X. Lu has shown asymptotic existence in v of resolvable BD(v, k, λ), yet for over twenty years the analogous problem for resolvable GD(v, G, λ) has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

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41

Nasiri, Ramin. "The reciprocal complementary Wiener number of graphs." Tamkang Journal of Mathematics 50, no. 4 (2019): 371–81. http://dx.doi.org/10.5556/j.tkjm.50.2019.2714.

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The reciprocal complementary Wiener number (RCW) of a connected graph G is defined as the sum ofweights frac{1}{D+1-d_G(x,y)} over all unordered vertex pairs in a graph G, where D is the diameter of Gand d_G(x,y) is the distance between vertices x and y. In this paper, we find new bounds for RCW ofgraphs, and study this invariant of two important types of graphs, named the Bar-Polyhex and theMycielskian graphs.

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42

K.Sunitha. "Radial Radio Pell Mean Labeling of Subdivision of Graphs." Advances in Nonlinear Variational Inequalities 28, no. 2 (2024): 267–73. http://dx.doi.org/10.52783/anvi.v28.1969.

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A one-one mapping ϕ : V (G) → N for a connected graph G is defined as follows: d(x,y)+⌈(ϕ(x)+2ϕ(y))/2⌉≥1+r(G), where radius is denoted by r(G). Any vertex in G has a radial radio pell mean number of ϕ which is the maximum number and is represented by rrpmn(ϕ). Here, we look at the labeling of various graphs using the radial radio pell mean of subdivisions such as subdivision of star graph S(K_(1,n)) , subdivision of path graph S(Pn) , subdivision of friendship graph S(Fn), subdivision of wheel graph S(Wn), subdivision of quadrilateral book graph S(QB(4,3)), subdivision of closed helm graph S(CHn), subdivision of helm graph S(Hn), subdivision of double fan graph S(DFn) and subdivision of triangular book graph S(TB(3,n)).

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43

Huda, Muhammad Nurul, and Hartono Hartono. "Metric Dimension of Banded-Turán Graph." PYTHAGORAS Jurnal Pendidikan Matematika 19, no. 1 (2024): 18–26. https://doi.org/10.21831/pythagoras.v19i1.68025.

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Given a graph G=(V(G),E(G)). Let S={s1,...,sk} be an ordered subset of V(G). Consider a vertex x, a coordinate of x with respect to S is represented as r(x|S)=(d(x,s1),...,d(x,sk)) where d(x,si) equals the number of edges in the shortest path between x and si for i=1,...,k. The minimum value of k such that for every x has distinct coordinate is called metric dimension of G. Turán graph T(n,r), n=r=2 is a subgraph of complete r-partite graph on n vertices having property that the difference of cardinality of any two distinct classes is at most one. In this paper, we build a new graph namely a banded-Turán graph, BT(n,r,m), as a graph built by a Turán graph T(n,r) and r uniform path graphs Pm in which every vertex of each class of T(n,r) connected to an initial vertex of corresponding path graph Pm. Intuitively, this graph illustrates as if a Turán graph is banded by r uniform ropes. We determine some basic graph properties including independence number, chromatic number, and diameter of banded-Turán. The main result in this paper is we obtain that the metric dimension of banded-Turán turns out that it depends on the number of its classes. If it has two classes then the metric dimension is equal to n-1 and if it has more than two classes then the metric dimension is equal to the metric dimension of Turán graph included in it.

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44

Rodríguez, José M., and José M. Sigarreta. "The hyperbolicity constant of infinite circulant graphs." Open Mathematics 15, no. 1 (2017): 800–814. http://dx.doi.org/10.1515/math-2017-0061.

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Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

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45

R, Stella Maragatham, and Subramanian A. "Results on Grundy Chromatic Number of Join Graph of Graphs." Ars Combinatoria 157 (December 31, 2023): 65–71. http://dx.doi.org/10.61091/ars157-06.

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A Grundy k -coloring of a graph G is a proper k -coloring of vertices in G using colors { 1 , 2 , ⋯ , k } such that for any two colors x and y , x < y , any vertex colored y is adjacent to some vertex colored x . The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph G , denoted by Γ ( G ) , is the largest integer k , such that there exists a Grundy k -coloring for G . It can be easily seen that Γ ( G ) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of G . In this paper, we obtain the Grundy chromatic number of Cartesian Product of path graph, complete graph, cycle graph, complete graph, wheel graph and star graph.

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46

Anantpiniwatna, Apinant, and Tiang Poomsa-Ard. "IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)." Asian-European Journal of Mathematics 02, no. 01 (2009): 1–17. http://dx.doi.org/10.1142/s1793557109000029.

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Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

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47

Chia, Gek-Ling, and Chong-Keang Lim. "Counting 2-circulant graphs." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 39, no. 2 (1985): 270–81. http://dx.doi.org/10.1017/s1446788700022527.

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AbstractAlspach and Sutcliffe call a graph X(S, q, F) 2-circulant if it consists of two isomorphic copies of circulant graphs X(p, S) and X(p, qS) on p vertices with “cross-edges” joining one another in a prescribed manner. In this paper, we enumerate the nonisomorphic classes of 2-circulant graphs X(S, q, F) such that |S| = m and |F| = k. We also determine a necessary and sufficient condition for a 2-circulant graph to be a GRR. The nonisomorphic classes of GRR on 2p vertices are also enumerated.

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48

LEFLOCH, PHILIPPE G. "GRAPH SOLUTIONS OF NONLINEAR HYPERBOLIC SYSTEMS." Journal of Hyperbolic Differential Equations 01, no. 04 (2004): 643–89. http://dx.doi.org/10.1142/s0219891604000287.

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For nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) we introduce a geometric framework in which solutions are sought as (continuous) parametrized graphs(t,s) ↦ (X,U)(t,s) satisfying ∂sX ≥ 0, rather than (discontinuous) functions (t,x) ↦ u(t,x). On one hand, we generalize an idea by Dal Maso, LeFloch, and Murat who used a family of traveling wave profiles to define non-conservative products, and we define the notion of graph solution subordinate to a family of Riemann graphs. The latter naturally encodes the graph of the solution to the Riemann problem, which should be determined from an augmented model taking into account small-scale physics and providing an internal structure to the shock waves. In a second definition, we write an evolution equation on the graphs directly and we introduce the notion of graph solution subordinate to a diffusion matrix, which merges together the hyperbolic equations (in the "non-vertical" parts of the graphs) with the traveling wave equation of the augmented model (in the "vertical" parts). We consider the Cauchy problem within the class of graph solutions. The graph solution to the Cauchy problem is constructed by completion of the discontinuities of the entropy solution. The uniqueness is established by applying a general uniqueness theorem due to Baiti, LeFloch, and Piccoli. The proposed geometric framework illustrates the importance of the uniform distance between graphs to deal with solutions of nonlinear hyperbolic problems.

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49

Aritonang, Winda, Nurdin Hinding, and Amir Kamal Amir. "Nilai Total Ketidakteraturan-H pada Graf Cn x P3." Jurnal Matematika, Statistika dan Komputasi 16, no. 1 (2019): 10. http://dx.doi.org/10.20956/jmsk.v16i1.5788.

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AbstrakPenentuan nilai total ketidakteraturan dari semua graf belum dapat dilakukan secara lengkap. Penelitian ini bertujuan untuk menentukan nilai total ketidakteraturan-H pada graf Cn x P3 untuk n ≥ 3 yang isomorfik dengan . Penentuan nilai total ketidakteraturan-H pada graf Cn x P3 dengan menentukan batas bawah terbesar dan batas atas terkecil. Batas bawah dianalisis berdasarkan sifat-sifat graf dan teorema pendukung lainnya. Sedangkan batas atas dianalisa dengan pemberian label pada titik dan sisi pada graf Cn x P3.Berdasarkan hasil penelitian ini diperoleh nilai total ketidakteraturan-H pada graf ths(Cn x P3, C4)=.Kata kunci : Selimut-H, Nilai total ketidakteraturan-HAbstractThe determine of H-irregularity total strength in all graphs was not complete on graph classes. The research aims to determine alghorithm the H-irregularity total strength of graph Cn x P3 for n ≥ 3 with use H-covering, where H is isomorphic to C4. The determine of H-irregularity total strength of graph Cn x P3 was conducted by determining lower bound and smallest upper bound. The lower bound was analyzed based on graph characteristics and other supporting theorem, while the upper bound was analyzed by edge labeling and vertex labeling of graph Cn x P3.The result show that the H-irregularity total strength of graph ths(Cn x P3, C4)=.Keyword : H-covering, H-irregularity total strength

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50

Ali, Jiwan Jalal, and Didar Abdulkhaleq Ali. "Stability and Maximum Independent Bond Set Polynomials of Painkiller Molecules Using Maximum Matching." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5992. https://doi.org/10.29020/nybg.ejpam.v18i2.5992.

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Chemical graph theory establishes a connection between the properties of molecules and their corresponding molecular graphs. A topological index is a graph invariant that characterizes the graph's structure and remains unaffected by graph automorphisms. In chemical graph theory, degree-based topological indices are particularly significant, offering crucial insights into the structural features of molecules. In this work, we introduce the maximum independent bond set polynomial $MIBSP(H;x,y)$, a powerful tool for deriving various degree-based topological indices. We specifically apply $MIBSP(H;x,y)$, to the chemical graphs of several painkiller molecules, including Aspirin, Paracetamol, Caffeine, Ibuprofen, Phenacetin, and Salicylic acid. The degree-based topological indices derived from these polynomials provide a deeper understanding of the molecular structures and their potential applications in pharmaceutical research.

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