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Journal articles on the topic 'Isolated vertices'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Chitra, P. Jeya Bala, and S. Arumugam. "Resolving Sets without Isolated Vertices." Procedia Computer Science 74 (2015): 38–42. http://dx.doi.org/10.1016/j.procs.2015.12.072.

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2

Enomoto, H. "Graph Decompositions without Isolated Vertices." Journal of Combinatorial Theory, Series B 63, no. 1 (1995): 111–24. http://dx.doi.org/10.1006/jctb.1995.1007.

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3

Dr., B. Ramireddy, Mohan Chand U., Sri Krishna Chaitanya A., and Sri Krishna Chaitanya A. "APPLICATIONS OF EDGE COLORING OF GRAPHS WITH RAINBOW NUMBERS PHENOMENA." International Journal of Research – Granthaalayah 3, no. 12 (2017): 163–70. https://doi.org/10.5281/zenodo.848967.

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This paper studies the Rainbow Ramsety Number for a non empty graph and the main results are 1. The Rainbow Ramsety Number of a graph F with out isolated vertices is defined if and only if F is a forest. 2. The Rainbow Ramsety Number of two graphs F1 and F2 with out isolated vertices is defined if and only if F1 is a star or F2 is a forest..

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4

ENOMOTO, Hikoe, and Shinsuke MATSUNAGA. "Graph decompositions without isolated vertices II." Journal of the Mathematical Society of Japan 49, no. 1 (1997): 161–80. http://dx.doi.org/10.2969/jmsj/04910161.

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5

Enomoto, Hikoe, and Shinsuke Matsunaga. "Graph decompositions without isolated vertices III." Journal of Graph Theory 24, no. 2 (1997): 155–64. http://dx.doi.org/10.1002/(sici)1097-0118(199702)24:2<155::aid-jgt3>3.0.co;2-p.

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6

Shang, Yilun. "On the Isolated Vertices and Connectivity in Random Intersection Graphs." International Journal of Combinatorics 2011 (May 30, 2011): 1–9. http://dx.doi.org/10.1155/2011/872703.

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We study isolated vertices and connectivity in the random intersection graph . A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When and , we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Rényi random graphs.

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7

Rajkhowa, Kukil Kalpa, and Helen K. Saikia. "Pendant and isolated vertices of comaximal graphs of modules." Boletim da Sociedade Paranaense de Matemática 41 (December 23, 2022): 1–8. http://dx.doi.org/10.5269/bspm.51488.

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A comaximal graph Γ(M) is an undirected graph with vertex set as the collection of all submodules of a module M and any two vertices A and B are adjacent if and only if A + B = M. We discuss characteristics of pendant vertices in Γ(M). We also observe features of isolated vertices in a special spanning subgraph in Γ(M).

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8

Chiba, Shuya, and Shinya Fujita. "Covering vertices by a specified number of disjoint cycles, edges and isolated vertices." Discrete Mathematics 313, no. 3 (2013): 269–77. http://dx.doi.org/10.1016/j.disc.2012.10.010.

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9

Rozario, J. Gerard, and J. Jon Arockiaraj. "Sum Labeling for Some Star and Cycle Related Special Graphs." Mapana - Journal of Sciences 11, no. 4 (2012): 77–90. http://dx.doi.org/10.12723/mjs.23.6.

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A sum labeling is a mapping from the vertices of G into the positive integers such that, for any two vertices u, v V (G) with labels (u) and (v), respectively, (uv) is an edge iff (u) + (v) is the label of another vertex in V (G). Any graph supporting such a labeling is called a sum graph. It is necessary to add (as a disjoint union) a component to sum label a graph. This disconnected component is a set of isolated vertices known as isolates and the labeling scheme that requires the fewest isolates is termed optimal. The number of isolates required for a graph to support a sum labeling is know

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10

Penrose, Mathew D. "Inhomogeneous random graphs, isolated vertices, and Poisson approximation." Journal of Applied Probability 55, no. 1 (2018): 112–36. http://dx.doi.org/10.1017/jpr.2018.9.

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Abstract Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Pois

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11

ALFARISI, RIDHO, DAFIK, ARIKA INDAH KRISTIANA, and IKA HESTI AGUSTIN. "Non-Isolated Resolving Number of Graph with Pendant Edges." Journal of Interconnection Networks 19, no. 02 (2019): 1950003. http://dx.doi.org/10.1142/s0219265919500038.

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We consider V, E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = {w1, w2, w3, …, wk} of vertices and a vertex v ∈ G, the ordered r(v|W) = (d(v, w1), d(v, w2), …, d(v, wk)) of k-vector is representations of v with respect to W, where d(v, w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of |W|. Furthermore, the resolving set W of graph G is called non-isolated resolving set

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12

Ramireddy, B., Mohan Chand, Sri Krishna Chaitanya, and B. R. Srinivas. "APPLICATIONS OF EDGE COLORING OF GRAPHS WITH RAINBOW NUMBERS PHENOMENA." International Journal of Research -GRANTHAALAYAH 3, no. 12 (2015): 163–70. http://dx.doi.org/10.29121/granthaalayah.v3.i12.2015.2901.

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This paper studies the Rainbow Ramsety Number for a non empty graph and the main results are 1. The Rainbow Ramsety Number of a graph F with out isolated vertices is defined if and only if F is a forest. 2. The Rainbow Ramsety Number of two graphs F1 and F2 with out isolated vertices is defined if and only if F1 is a star or F2 is a forest.

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13

Huang, Shaobin, Jiang Zhou, and Changjiang Bu. "Signless Laplacian spectral characterization of graphs with isolated vertices." Filomat 30, no. 14 (2016): 3689–96. http://dx.doi.org/10.2298/fil1614689h.

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A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G ? rK1 is DQS under certain conditions. Applying these results, some DQS graphs with isolated vertices are obtained.

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14

Boyar, Joan, and Christian Kudahl. "Adding isolated vertices makes some greedy online algorithms optimal." Discrete Applied Mathematics 246 (September 2018): 12–21. http://dx.doi.org/10.1016/j.dam.2017.02.025.

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15

BEREZNY, STEFAN, and MICHAL STAS. "Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six." Carpathian Journal of Mathematics 35, no. 2 (2019): 137–46. http://dx.doi.org/10.37193/cjm.2019.02.02.

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The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the

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16

Jent, Emma, and Ping Zhang. "The Vertex-Disjoint and Edge-Disjoint Ramsey Numbers of a Set of Graphs." Axioms 14, no. 7 (2025): 486. https://doi.org/10.3390/axioms14070486.

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The Ramsey number R(F) of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of Kn produces a subgraph isomorphic to F all of whose edges are colored the same. Let F be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number VRt(F) is the smallest positive integer n such that every red–blue coloring of the complete graph Kn of order n results in at least t pairwise vertex-disjoint monochromatic graphs in F; while the edge-disjoint Ramsey number ERt(F) is the smallest positive integer n such tha

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17

Al-Hasan, Z. A., and M. A. Al-Mselih. "The topological space generated from the graphs by use of some relation." Journal of Interdisciplinary Mathematics 26, no. 4 (2023): 613–17. http://dx.doi.org/10.47974/jim-1479.

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In this article, we study the topology and its properties generated from a simple undirected graph with no isolated vertices. We use a simple undirected graph to generate the topology, which connects event edges with the neighboring relationship between vertices, edges, and studies some properties of the graph.

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18

Gutman, Ivan, and Peter Paule. "The variance of the vertex degrees of randomly generated graphs." Publikacije Elektrotehnickog fakulteta - serija: matematika, no. 13 (2002): 30–35. http://dx.doi.org/10.2298/petf0213030g.

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We consider graphs with n vertices and m edges constructed from n isolated vertices by selecting among them, uniformly at random, m pairs and connecting them by m edges. The variance of the vertex degrees of such graphs is shown to be equal to 2m (n - n ? 2m)/(n 3+ n ) . In order to arrive at this formula some combinatorial identities are verified.

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19

Lo, Allan, and Nicolás Sanhueza-Matamala. "An asymptotic bound for the strong chromatic number." Combinatorics, Probability and Computing 28, no. 5 (2019): 768–76. http://dx.doi.org/10.1017/s0963548318000561.

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AbstractThe strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding $r\lceil n/r\rceil-n$ isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c > 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

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20

Reji, T., and Jinitha Varughese. "Least common multiple of graphs." Discrete Mathematics, Algorithms and Applications 08, no. 02 (2016): 1650032. http://dx.doi.org/10.1142/s1793830916500324.

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A graph [Formula: see text] without isolated vertices is said to be a least common multiple of two graphs [Formula: see text] and [Formula: see text] if [Formula: see text] is a graph of minimum size such that [Formula: see text] is both [Formula: see text] decomposable and [Formula: see text] decomposable. Chartrand et al. proved that every two non-empty graphs [Formula: see text] and [Formula: see text] without isolated vertices have a least common multiple. Size of a least common multiple of [Formula: see text] and [Formula: see text] is denoted by [Formula: see text]. In this paper, least

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21

BEREŽNÝ, ŠTEFAN, and MICHAL STAŠ. "On the crossing numbers of the join products of five graphs on six vertices with discrete graph." Carpathian Journal of Mathematics 39, no. 2 (2022): 371–82. http://dx.doi.org/10.37193/cjm.2023.02.03.

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The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, the crossing number of the join product $G^\ast + D_n$ for the connected graph $G^\ast$ on six vertices consisting of one path on four vertices $P_4$ and two leaves adjacent with the same outer vertex of the path $P_4$ is given, where $D_n$ consists of $n$ isolated vertices. Finally, by adding some edges to the graph $G^\ast$, we obtain the crossing numbers of the join products of other four graphs with $D_n$.

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22

DOBSON, EDWARD. "Packing Trees into the Complete Graph." Combinatorics, Probability and Computing 11, no. 3 (2002): 263–72. http://dx.doi.org/10.1017/s0963548301005077.

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Let c [les ] 0.076122 and T1, T2,…, Tn be a sequence of trees such that [mid ]V(Ti)[mid ] [les ] i−c(i−1). We prove that, if for each 1 [les ] i [les ] n there exists a vertex xi ∈ V(Ti) such that Ti−xi has at least (1−2c)(i−1) isolated vertices, then T1,…, Tn can be packed into Kn. We also prove that if T is a tree of order n+1−c′n, c′ [les ] 1/25 (37−8 √21 ) ≈ 0.0135748, such that there exists a vertex x ∈ V(T) and T−x has at least n(1−2c′) isolated vertices, then 2n+1 copies of T may be packed into K2n+1.

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23

CRUZ, ROBERTO, and JUAN RADA. "Extremal Graphs for Exponential VDB Indices." Kragujevac Journal of Mathematics 46, no. 1 (2022): 105–13. http://dx.doi.org/10.46793/kgjmat2201.105c.

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24

Puhalskii, Anatolii A. "On the number of isolated vertices in a growing random graph." Rocky Mountain Journal of Mathematics 43, no. 6 (2013): 1941–89. http://dx.doi.org/10.1216/rmj-2013-43-6-1941.

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25

Kaneko, Atsushi, Ken-ichi Kawarabayashi, Katsuhiro Ota, and Kiyoshi Yoshimoto. "On a hamiltonian cycle in which specified vertices are not isolated." Discrete Mathematics 258, no. 1-3 (2002): 85–91. http://dx.doi.org/10.1016/s0012-365x(02)00263-7.

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26

Al-Harere, Manal N., and Mohammed A. Abdlhusein. "Pitchfork domination in graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (2020): 2050025. http://dx.doi.org/10.1142/s1793830920500251.

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In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pit

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27

Staš, Michal. "Parity Properties of Configurations." Mathematics 10, no. 12 (2022): 1998. http://dx.doi.org/10.3390/math10121998.

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In the paper, the crossing number of the join product G*+Dn for the disconnected graph G* consisting of two components isomorphic to K2 and K3 is given, where Dn consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of G*. For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of G* with paths Pn and cycles Cn on n

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28

Shang, Yilun. "Super Connectivity of Erdős-Rényi Graphs." Mathematics 7, no. 3 (2019): 267. http://dx.doi.org/10.3390/math7030267.

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The super connectivity κ ′ ( G ) of a graph G is the minimum cardinality of vertices, if any, whose deletion results in a disconnected graph that contains no isolated vertex. G is said to be r-super connected if κ ′ ( G ) ≥ r . In this note, we establish some asymptotic almost sure results on r-super connectedness for classical Erdős–Rényi random graphs as the number of nodes tends to infinity. The known results for r-connectedness are extended to r-super connectedness by pairing off vertices and estimating the probability of disconnecting the graph that one gets by identifying the two vertice

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29

Priyadharshini, V., and M. Nalliah. "Local distance antimagic chromatic number for the union of star and double star graphs." Ukrains’kyi Matematychnyi Zhurnal 75, no. 5 (2023): 669–82. http://dx.doi.org/10.37863/umzh.v75i5.7075.

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UDC 519.17 Let G = ( V , E ) be a graph on p vertices with no isolated vertices. A bijection f from V to { 1,2 ,3 , … , p } is called a local distance antimagic labeling if, for any two adjacent vertices u and v , we receive distinct weights (colors), where a vertex x has the weight w ( x ) = ∑ v ϵ N ( x ) f ( v ) . The local distance antimagic chromatic number χ l ⅆ a ( G ) is defined as the least number of colors used in any local distance antimagic labeling of G . We determine the local distance antimagic chromatic number for the disjoint union of t copies of stars and double stars.

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30

M, Nalliah. "Local distance antimagic chromatic number for the union of complete bipartite graphs." Tamkang Journal of Mathematics 54, no. 4 (2022): 281–91. http://dx.doi.org/10.5556/j.tkjm.54.2023.4804.

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Let G be a graph on p vertices and q edges with no isolated vertices. A bijection f : V → {1, 2, 3, ..., p} is called local distance antimagic labeling, if for any two adjacent vertices u and v, we have w(u) is not equal to w(v), where w(u) is the sum of all neighbour labels of u. The local distance antimagic chromatic number χlda(G) is defined to be the minimum number of colors taken overall colorings of G induced by local distance antimagic labelings of G. In this paper, we determine the graph G for the local distance antimagic chromatic number is 2.

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31

L. Hadi, Zahraa, and Asmhan F. Hassan. "The Non-Adjacency Incompatible Vertices Topology on Digraphs." Al-Nahrain Journal of Science 27, no. 4 (2024): 28–33. https://doi.org/10.22401/anjs.27.4.05.

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In this work, we introduce a new type of topology associated to directed graph on vertices set, which is named as non-adjacency incompatible vertices topology (𝑁𝐼𝑉-topology) of directed graph. A sub-basis family of this topology is generated on the set of vertices, and formed via taking the non-adjacency vertices which arises (path of length two) in different direction to each vertex. We investigate some properties and discuss it on an important and certain types of directed graphs. Our motivation is to giving fundamental steps toward investigation of various properties of directed graphs via

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32

Kolotoglu, Emre. "A complete solution to the spectrum problem for graphs with six vertices and up to nine edges." Contributions to Discrete Mathematics 15, no. 3 (2020): 1–24. http://dx.doi.org/10.55016/ojs/cdm.v15i3.62290.

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Let $G$ be a graph. A $G$-design of order $n$ is a decomposition of the complete graph $K_n$ into disjoint copies of $G$. The existence problem of graph designs has been completely solved for all graphs with up to five vertices, and all graphs with six vertices and up to seven edges; and almost completely solved for all graphs with six vertices and eight edges leaving two cases of order 32 unsettled. Up to isomorphism there are 20 graphs with six vertices and nine edges (and no isolated vertex). The spectrum problem has been solved completely for 11 of these graphs, and partially for 2 of thes

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33

K. Chinju Krishna, C. David Raj, and K. Rubin Mary. "Some standard results on even sum graphs." Malaya Journal of Matematik 8, no. 04 (2020): 2282–84. http://dx.doi.org/10.26637/mjm0804/0164.

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A sum graph is a graph for which there is a labeling of its vertices with distinct positive integers so that two vertices are adjacent if and only if the sum of their labels is the label of another vertex. Integral sum graphs are defined similarly, except that the labels may be any integers. The concept of Even Sum Graphs was introduced by C. David Raj, et al. A graph \(G\) is called an even sum graph if there is a labeling \(\theta\) of its vertices with distinct non - negative even integers, so that for any two distinct vertices \(u\) and \(v, u v\) is an edge of \(G\) if and only if \(\thet

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34

STAŠ, MICHAL. "ON THE CROSSING NUMBER OF THE JOIN OF THE WHEEL ON FIVE VERTICES WITH THE DISCRETE GRAPH." Bulletin of the Australian Mathematical Society 101, no. 3 (2019): 353–61. http://dx.doi.org/10.1017/s0004972719001199.

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We give the crossing number of the join product $W_{4}+D_{n}$, where $W_{4}$ is the wheel on five vertices and $D_{n}$ consists of $n$ isolated vertices. The proof is based on calculating the minimum number of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph $W_{4}$ and from the set of subgraphs which cross the edges of $W_{4}$ exactly once.

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35

GIUDICI, MICHAEL, and BOJAN KUZMA. "REALIZABILITY PROBLEM FOR COMMUTING GRAPHS." Journal of the Australian Mathematical Society 101, no. 3 (2016): 335–55. http://dx.doi.org/10.1017/s1446788716000148.

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We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

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36

Garzoni, Daniele. "The invariably generating graph of the alternating and symmetric groups." Journal of Group Theory 23, no. 6 (2020): 1081–102. http://dx.doi.org/10.1515/jgth-2019-0187.

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AbstractGiven a finite group G, the invariably generating graph of G is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of G, and two classes are adjacent if and only if they invariably generate G. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most 6.

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37

WANG, CHUNXIA, YALAN LI, SHUMIN ZHANG, CHENGFU YE, and XIA WANG. "Matching Preclusion Number of Radix Triangular Mesh with Odd Vertices." Journal of Interconnection Networks 18, no. 02n03 (2018): 1850010. http://dx.doi.org/10.1142/s021926591850010x.

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The matching preclusion number of graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or almost-perfect matching. The strong matching preclusion number of a graph G is the minimum number of vertices and edges whose deletion leaves the resulting graph without a perfect matching or an almost-perfect matching. The conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or almost-perfect matching. In this paper, we study the m

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38

STAŠ, MICHAL. "DETERMINING CROSSING NUMBERS OF GRAPHS OF ORDER SIX USING CYCLIC PERMUTATIONS." Bulletin of the Australian Mathematical Society 98, no. 3 (2018): 353–62. http://dx.doi.org/10.1017/s000497271800059x.

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We extend known results concerning crossing numbers by giving the crossing number of the join product$G+D_{n}$, where the connected graph$G$consists of one$4$-cycle and of two leaves incident with the same vertex of the$4$-cycle, and$D_{n}$consists of$n$isolated vertices. The proofs are done with the help of software that generates all cyclic permutations for a given number$k$and creates a graph for calculating the distances between all$(k-1)!$vertices of the graph.

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39

Staš, Michal. "Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five." Symmetry 11, no. 2 (2019): 123. http://dx.doi.org/10.3390/sym11020123.

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The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing num

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40

Vaidya, S. K., and A. D. Parmar. "Some More Results on Total Equitable Bondage Number of A Graph." Journal of Scientific Research 11, no. 3 (2019): 303–9. http://dx.doi.org/10.3329/jsr.v11i3.40573.

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The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G-E0) > γ (G). An equitable dominating set D is called a total equitable dominating set if the induced subgraph < D > has no isolated vertices. The total equitable domination number γte(G) of G is the minimum cardinality of a total equitable dominating set of G. If γte(G) ≠ |V(G)| and <G-E0> contains no isolated vertices then the total equitable bondage number bte(G) of a graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γte(G-E0) &g

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41

Ding, Zongpeng, and YuanQiu Huang. "The crossing numbers of join of some graphs with n isolated vertices." Discussiones Mathematicae Graph Theory 38, no. 4 (2018): 899. http://dx.doi.org/10.7151/dmgt.2048.

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42

Bu, Changjiang, Jiang Zhou, Hongbo Li, and Wenzhe Wang. "Spectral Characterizations of the Corona of a Cycle and Two Isolated Vertices." Graphs and Combinatorics 30, no. 5 (2013): 1123–33. http://dx.doi.org/10.1007/s00373-013-1327-7.

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Bender, Edward A., E. Rodney Canfield, and Brendan D. McKay. "The Asymptotic Number of Labeled Graphs withnVertices,qEdges, and No Isolated Vertices." Journal of Combinatorial Theory, Series A 80, no. 1 (1997): 124–50. http://dx.doi.org/10.1006/jcta.1997.2798.

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Varghese, Eddith Sarah, D. Antony Xavier, Ammar Alsinai, Deepa Mathew, S. Arul Amirtha Raja, and Hanan Ahmed. "Strong Total Monophonic Problems in Product Graphs, Networks, and Its Computational Complexity." Journal of Mathematics 2022 (September 8, 2022): 1–7. http://dx.doi.org/10.1155/2022/6194734.

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Let G be a graph with vertex set as V G and edge set as E G which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T ⊆ V G , which contains no isolated vertices, and all the vertices in V G \ T lie on a fixed unique chordless path between the pair of vertices in T . The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by s m t G . We proved the NP-completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and network

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Nalliah, M. "On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs." Proyecciones (Antofagasta) 43, no. 2 (2024): 473–94. http://dx.doi.org/10.22199/issn.0717-6279-5963.

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Let $G$ be a graph on $p$ vertices and $q$ edges with no isolated vertices. A bijection $f: V\rightarrow \{1,2,3,...,p\}$ is called local distance antimagic labeling, if for any two adjacent vertices $u$ and $v$, we have $w(u) \neq w(v)$, where $w(u)=\sum_{x\epsilon N(u)} {f(x)}$. The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings of $G$. In this paper, we obtained the necessary and sufficient condition for the local distance antimagic chromatic number of some

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Sheikholeslami, Seyed Mahmoud, and Lutz Volkmann. "Outer independent total double Italian domination number." Computer Science Journal of Moldova 32, no. 1(94) (2024): 19–37. http://dx.doi.org/10.56415/csjm.v32.02.

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If $G$ is a graph with vertex set $V(G)$, then let $N[u]$ be the closed neighborhood of the vertex $u\in V(G)$. A total double Italian dominating function (TDIDF) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2,3\}$ satisfying (i) $f(N[u])\ge 3$ for every vertex $u\in V(G)$ with $f(u)\in\{0,1\}$ and (ii) the subgraph induced by the vertices with a non-zero label has no isolated vertices. A TDIDF is an outer-independent total double Italian dominating function (OITDIDF) on $G$ if the set of vertices labeled $0$ induces an edgeless subgraph. The weight of an OITDIDF is the sum of its func

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C. Jayasekaran and Binoja L G. "Relatively prime restrained detour domination number of a graph." Gulf Journal of Mathematics 16, no. 2 (2024): 291–97. http://dx.doi.org/10.56947/gjom.v16i2.1844.

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Consider a connected graph G=(V, E) with a minimum of two vertices. A subset ⊆ is termed a relatively prime restrained detour dominating set of G if it fulfills two conditions: firstly, S must be a relatively prime detour dominating set, and secondly, the induced subgraph should not contain any isolated vertices. The relatively prime restrained detour domination number, denoted as γrprdn(G), is defined as the minimum cardinality of such a set that satisfies these conditions. Precise values for certain standard graphs, limits and some interesting results are established.

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DUNCAN, ANDREW J., and VLADIMIR N. REMESLENNIKOV. "AUTOMORPHISMS OF PARTIALLY COMMUTATIVE GROUPS II: COMBINATORIAL SUBGROUPS." International Journal of Algebra and Computation 22, no. 07 (2012): 1250074. http://dx.doi.org/10.1142/s0218196712500749.

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We define several "standard" subgroups of the automorphism group Aut (G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut (G). If C is the commutation graph of G, we show how Aut (G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decomposition of Aut (G) into a subgroup of locally conjugating automorphisms by the subgroup stabilizing a

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Parveen and Jitender Kumar. "The Complement of Enhanced Power Graph of a Finite Group." Algebra Colloquium 32, no. 01 (2025): 111–28. https://doi.org/10.1142/s1005386725000100.

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The enhanced power graph [Formula: see text] of a finite group [Formula: see text] is the simple undirected graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if [Formula: see text], [Formula: see text] for some [Formula: see text]. We give an affirmative answer to the question posed by Cameron which asks whether the complement of the enhanced power graph [Formula: see text] of a non-cyclic group[Formula: see text], [Formula: see text], has only one connected component apart from isolated vertices. We classify all finit

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Johnson, Aidan, Andrew E. Vick, and Darren A. Narayan. "Characterization of All Graphs with a Failed Skew Zero Forcing Number of 1." Mathematics 10, no. 23 (2022): 4463. http://dx.doi.org/10.3390/math10234463.

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Given a graph G, the zero forcing number of G, Z(G), is the minimum cardinality of any set S of vertices of which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Hence the failed zero forcing number of a graph was defined to be the cardinality of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor u of v is not in

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