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Journal articles on the topic 'Boundy, K A V'

Author: Grafiati

Published: 10 December 2022

Last updated: 19 July 2025

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1

Baicheva, Tsonka, and Svetlana Topalova. "Maximal (v, k, 2, 1) Optical Orthogonal Codes with k = 6 and 7 and Small Lengths." Mathematics 11, no. 11 (2023): 2457. http://dx.doi.org/10.3390/math11112457.

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Optical orthogonal codes (OOCs) are used in optical code division multiple access systems to allow a large number of users to communicate simultaneously with a low error probability. The number of simultaneous users is at most as big as the number of codewords of such a code. We consider (v,k,2,1)-OOCs, namely OOCs with length v, weight k, auto-correlation 2, and cross-correlation 1. An upper bound B0(v,k,2,1) on the maximal number of codewords of such an OOC was derived in 1995. The number of codes that meet this bound, however, is very small. For k≤5, the (v,k,2,1)-OOCs have already been tho

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2

Zhang, Xinhong, Xin Song, and Ruijuan Li. "[k]-Roman Domination in Digraphs." Symmetry 15, no. 3 (2023): 743. http://dx.doi.org/10.3390/sym15030743.

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Let D=(V(D),A(D)) be a finite, simple digraph and k a positive integer. A function f:V(D)→{0,1,2,…,k+1} is called a [k]-Roman dominating function (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} and AN−[v]=AN−(v)∪{v}. The weight of a [k]-RDF f is ω(f)=∑v∈V(D)f(v). The minimum weight of any [k]-RDF on D is the [k]-Roman domination number, denoted by γ[kR](D). For k=2 and k=3, we call them the double Roman domination number and the triple Roman domination number, respectively. In this paper, we presented some general bounds and the Nordhaus–Gaddum

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3

Cabrera-Martínez, Abel. "Total outer-independent domination in regular graphs." Filomat 38, no. 18 (2024): 6581–86. https://doi.org/10.2298/fil2418581c.

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Let G be a connected graph of order n. A set D ? V(G) is a total outer-independent dominating set of G if N(v) ? D , ? for every v ? D and N(v) ? D for every v ? V(G)\D. The total outer-independent domination number of G, denoted by ?oi t (G), is the minimum cardinality among all total outer-independent dominating sets of G. We show that if G is a k-regular graph with k ? 3, then (k/2k?1) n ? ?oi t (G) ? (k k+1) n. In addition, we characterize the k-regular graphs satisfying the above bounds, except for the case of cubic graphs attaining the upper bound. Finally, we obtain improved bounds (wit

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Fleming-Vázquez, Nicholas. "Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-Mixing Nonuniformly Hyperbolic Maps." Communications in Mathematical Physics 391, no. 1 (2022): 173–98. http://dx.doi.org/10.1007/s00220-022-04325-w.

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AbstractConsider a nonuniformly hyperbolic map $$ T:M\rightarrow M $$ T : M → M modelled by a Young tower with tails of the form $$ O(n^{-\beta }) $$ O ( n - β ) , $$ \beta >2 $$ β > 2 . We prove optimal moment bounds for Birkhoff sums $$ \sum _{i=0}^{n-1}v\circ T^i $$ ∑ i = 0 n - 1 v ∘ T i and iterated sums $$ \sum _{0\le i<j<n}v\circ T^i\, w\circ T^j $$ ∑ 0 ≤ i < j < n v ∘ T i w ∘ T j , where $$ v,w:M\rightarrow {{\mathbb {R}}} $$ v , w : M → R are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $$ \beta >5$$ β > 5 . Our method

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Weigel, A. Gambini, and T. S. Weigel. "On the orders of primitive linear P'-groups." Bulletin of the Australian Mathematical Society 48, no. 3 (1993): 495–521. http://dx.doi.org/10.1017/s0004972700015951.

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A group G ≤ GLK(V) is called K-primitive if there exists no non-trivial decomposition of V into a sum of K-spaces which is stabilised by G. We show that if V is a finite vector space and G a K-primitive subgroup of GLK(V) whose order is coprime to |V|, we can bound the order of G by |V|log2(|V|) apart from one exception. Later we use this result to obtain some lower bounds on the number of p–singular elements in terms of the group order and the minimal representation degree.

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Hernández Mira, Frank A., Ernesto Parra Inza, José M. Sigarreta Almira, and Nodari Vakhania. "Properties of the Global Total k-Domination Number." Mathematics 9, no. 5 (2021): 480. http://dx.doi.org/10.3390/math9050480.

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A nonempty subset D⊂V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D⊆V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v∈V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G¯. The global total k-domination number of G, denoted by γktg(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γktg(G), and dev

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SHEIKHOLESLAMI, S. M., and L. VOLKMANN. "SIGNED TOTAL {K}-DOMINATION AND {K}-DOMATIC NUMBERS OF GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 01 (2012): 1250006. http://dx.doi.org/10.1142/s1793830912500061.

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Let k be a positive integer, and let G be a simple graph with vertex set V(G). A function f : V(G) → {±1, ±2, …, ±k} is called a signed total {k}-dominating function if ∑u∈N(v) f(u) ≥ k for each vertex v ∈ V(G). A set {f1, f2, …, fd} of signed total {k}-dominating functions on G with the property that [Formula: see text] for each v∈V(G), is called a signed total {k}-dominating family (of functions) on G. The maximum number of functions in a signed total {k}-dominating family on G is the signed total {k}-domatic number of G, denoted by [Formula: see text]. Note that [Formula: see text] is the c

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8

Yang, Hong, Pu Wu, Sakineh Nazari-Moghaddam, et al. "Bounds for signed double Roman k-domination in trees." RAIRO - Operations Research 53, no. 2 (2019): 627–43. http://dx.doi.org/10.1051/ro/2018043.

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Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this

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Bouchou, Ahmed, Mustapha Chellali, and Mohamed Zemir. "On independent [1, k]-set in graphs." Filomat 38, no. 30 (2024): 10687–94. https://doi.org/10.2298/fil2430687b.

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For an integer k ? 1, a subset S ? V in a graph G = (V, E) is an independent [1, k]-set of G if S is independent and every vertex in V-S is adjacent to one but no more than k vertices in S. The upper [1, k]-independence number noted ?[1,k] (G) is the maximum cardinality of an independent [1, k]-set of G. In this paper, we provide a constructive characterization of graphs having an independent [1, k]-set, while for split graphs, a necessary and sufficient condition is given for those having an independent [1, k]-set. Moreover, some upper bounds o[1,k]n ? (G) are established for graphs having an

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Dehgardi, Nasrin, Maryam Atapour, and Abdollah Khodkar. "Twin signed k-domination numbers in directed graphs." Filomat 31, no. 20 (2017): 6367–78. http://dx.doi.org/10.2298/fil1720367d.

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Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers

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Kim, Kijung. "On k-rainbow domination in middle graphs." RAIRO - Operations Research 55, no. 6 (2021): 3447–58. http://dx.doi.org/10.1051/ro/2021163.

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Let G be a finite simple graph with vertex set V(G) and edge set E(G). A function f : V(G) → P({1,2,…,k}) is a k-rainbow dominating function on G if for each vertex v∈V(G) for which f(v) = ∅, it holds that ⋃u∈N(v) f(u) = {1,2,…,k}. The weight of a k-rainbow dominating function is the value ∑v∈V(G)|f(v)|. The k-rainbow domination number γrk (G) is the minimum weight of a k-rainbow dominating function on G. In this paper, we initiate the study of k-rainbow domination numbers in middle graphs. We define the concept of a middle k-rainbow dominating function, obtain some bounds related to it and de

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Koam, Ali N. A., Ali Ahmad, Martin Bača, and Andrea Semaničová-Feňovčíková. "Modular edge irregularity strength of graphs." AIMS Mathematics 8, no. 1 (2022): 1475–87. http://dx.doi.org/10.3934/math.2023074.

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<abstract><p>For a simple graph $ G = (V, E) $ with the vertex set $ V(G) $ and the edge set $ E(G) $, a vertex labeling $ \varphi: V(G) \to \{1, 2, \dots, k\} $ is called a $ k $-labeling. The weight of an edge under the vertex labeling $ \varphi $ is the sum of the labels of its end vertices and the modular edge-weight is the remainder of the division of this sum by $ |E(G)| $. A vertex $ k $-labeling is called a modular edge irregular if for every two different edges their modular edge-weights are different. The maximal integer $ k $ minimized over all modular edge irregular $ k

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Kuziak, Dorota, Iztok Peterin, and Ismael Yero. "Computing the (k-)monopoly number of direct product of graphs." Filomat 29, no. 5 (2015): 1163–71. http://dx.doi.org/10.2298/fil1505163k.

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Let G = (V,E) be a simple graph without isolated vertices and minimum degree ?(G), and let k ? {1-??(G)/2? ,..., ?(G)/2c?} be an integer. Given a set M ? V, a vertex v of G is said to be k-controlled by M if ?M(v)? ?(v)/2 + k where ?M(v) represents the quantity of neighbors v has in M and ?(v) the degree of v. The set M is called a k-monopoly if it k-controls every vertex v of G. The minimum cardinality of any k-monopoly is the k-monopoly number of G. In this article we study the k-monopoly number of direct product graphs. Specifically we obtain tight lower and upper bounds for the k-monopoly

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Caadan, Julius Guhiting, Rolando N. Paluga, and Imelda S. Aniversario. "Upper Distance k-Cost Effective Number in the Join of Graphs." European Journal of Pure and Applied Mathematics 13, no. 3 (2020): 701–9. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3657.

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Let k be a positive integer and G be a connected graph. The open k-neighborhoodset Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v} : dG(u, v) ≤ k}. A set S of vertices of G is a distance k- cost effective if for every vertex u in S, |Nk G(u) ∩ Sc| − |NkG(u) ∩ S| ≥ 0. The maximum cardinality of a distance k- cost effective set of G is called the upper distance k- cost effective number of G. In this paper, we characterized a distance k- cost effective set in the join of two graphs. As direct consequences, the bounds or the exact values of the upper distance k- cost effective numbers a

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Harant, Jochen, and Stanislav Jendrol'. "Lightweight paths in graphs." Opuscula Mathematica 39, no. 6 (2019): 829–37. http://dx.doi.org/10.7494/opmath.2019.39.6.829.

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Let \(k\) be a positive integer, \(G\) be a graph on \(V(G)\) containing a path on \(k\) vertices, and \(w\) be a weight function assigning each vertex \(v\in V(G)\) a real weight \(w(v)\). Upper bounds on the weight \(w(P)=\sum_{v\in V(P)}w(v)\) of \(P\) are presented, where \(P\) is chosen among all paths of \(G\) on \(k\) vertices with smallest weight.

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Cheng, Rui, Gohar Ali, Gul Rahmat, Muhammad Yasin Khan, Andrea Semanicova-Fenovcikova, and Jia-Bao Liu. "Investigation of General Power Sum-Connectivity Index for Some Classes of Extremal Graphs." Complexity 2021 (August 20, 2021): 1–8. http://dx.doi.org/10.1155/2021/6623277.

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In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower

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Caadan, Julius Guhiting, Rolando N. Paluga, and Imelda S. Aniversario. "Distance k-Cost Effective Sets in the Corona and Lexicographic Product of Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 261–70. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4381.

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Let G be a connected graph and k ≥ 1 be an integer. The open k-neighborhood Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v}: dG(u, v) ≤ k}. A set S of vertices of G is called distance k-cost effective of G if for every vertex u in S, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| ≥ 0. The maximum cardinality of a distance k-cost effective set of G is called the upper distance k-cost effective number of G. In this paper, we characterized the distance k-cost effective sets in the corona and lexicographic product of two graphs. Consequently, the bounds or the exact values of the upper distanc

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18

Shao, Zehui, Rija Erveš, Huiqin Jiang, Aljoša Peperko, Pu Wu, and Janez Žerovnik. "Double Roman Graphs in P(3k, k)." Mathematics 9, no. 4 (2021): 336. http://dx.doi.org/10.3390/math9040336.

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A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman

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Liu, Hongfang, Jinxia Liang, Yuhu Liu, and Kinkar Chandra Das. "A Combinatorial Approach to Study the Nordhaus–Guddum-Type Results for Steiner Degree Distance." Mathematics 11, no. 3 (2023): 738. http://dx.doi.org/10.3390/math11030738.

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In 1994, Dobrynin and Kochetova introduced the concept of degree distance DD(Γ) of a connected graph Γ. Let dΓ(S) be the Steiner k-distance of S⊆V(Γ). The Steiner Wiener k-index or k-center Steiner Wiener indexSWk(Γ) of Γ is defined by SWk(Γ)=∑|S|=kS⊆V(Γ)dΓ(S). The k-center Steiner degree distanceSDDk(Γ) of a connected graph Γ is defined by SDDk(Γ)=∑|S|=kS⊆V(Γ)∑v∈SdegΓ(v)dΓ(S), where degΓ(v) is the degree of the vertex v in Γ. In this paper, we consider the Nordhaus–Gaddum-type results for SWk(Γ) and SDDk(Γ). Upper bounds on SWk(Γ)+SWk(Γ¯) and SWk(Γ)·SWk(Γ¯) are obtained for a connected graph

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Aghigh, Kamal, and Sudesh K. Khanduja. "ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD." Proceedings of the Edinburgh Mathematical Society 45, no. 1 (2002): 219–27. http://dx.doi.org/10.1017/s0013091500000936.

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AbstractLet $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred

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Girish, Lakshmi, and Kanagasabapathi Somasundaram. "Bound for the k-Fault-Tolerant Power-Domination Number." Symmetry 16, no. 7 (2024): 781. http://dx.doi.org/10.3390/sym16070781.

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A set S⊆V is referred to as a k-fault-tolerant power-dominating set of a given graph G=(V,E) if the difference S∖F remains a power-dominating set of G for any F⊆S with |F|≤k, where k is an integer with 0≤k<|V|. The lowest cardinality of a k-fault-tolerant power-dominating set is the k-fault-tolerant power-domination number of G, denoted by γPk(G). Generalized Petersen graphs GP(m,k) and generalized cylinders SG are two well-known graph classes. In this paper, we calculate the k-fault-tolerant power-domination number of the generalized Petersen graphs GP(m,1) and GP(m,2). Also, we obtain γPk

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Petrosyan, Petros A., and Levon N. Muradyan. "SOME BOUNDS ON THE NUMBER OF COLORS IN INTERVAL EDGE-COLORINGS OF GRAPHS." Proceedings of the YSU A: Physical and Mathematical Sciences 58, no. 2 (264) (2024): 57–65. https://doi.org/10.46991/pysua.2024.58.2.057.

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An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval \lb $t$-coloring}, if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A vertex $v$ of a graph $G=(V,E)$ is called a dominating vertex if $d_{G}(v)=|V|-1$, where $d_{G}(v)$ is the degree of $v$ in $G$. In this paper we prove, that if $G$ is a graph with the dominating vertex $u$ and it has an interval $t$-coloring, then $t\leq |V|+2\Delta(G-u)-1$, where $\Delta(G)$ is the maximum degree of $G$. We also show, that if a $k$-connected graph

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Volkmann, Lutz. "Signed total k-independence in digraphs." Filomat 28, no. 10 (2014): 2121–30. http://dx.doi.org/10.2298/fil1410121v.

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Let k ? 2 be an integer. A function f:V(D) ? {-1,1} defined on the vertex set V(D) of a digraph D is a signed total k-independence function if ?x?N-(v)f(x) ? k - 1 for each v ? V(D), where N-(v) consists of all vertices of D from which arcs go into v. The weight of a signed total k-independence function f is defined by w(f)=?x?V(D)f(x). The maximum of weights w(f), taken over all signed total k-independence functions f on D, is the signed total k-independence number k?st(D) of D. In this work, we mainly present upper bounds on k?st(D), as for example k?st(D) ? n-2? ?- + 1-k)/2? and k?st(D)? ?+

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Ghanem, Manal, Hasan Al-Ezeh, and Ala’a Dabbour. "Locating Chromatic Number of Powers of Paths and Cycles." Symmetry 11, no. 3 (2019): 389. http://dx.doi.org/10.3390/sym11030389.

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Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring wit

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Beidar, Konstantin I., and Robert Wisbauer. "On uniform bounds of primeness in matrix rings." Journal of the Australian Mathematical Society 76, no. 2 (2004): 167–74. http://dx.doi.org/10.1017/s1446788700008879.

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AbstractA subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices

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Agustín-Aquino, Octavio A. "A note on a sumset in \(\mathbb{Z}_{2k}\)." Online Journal of Analytic Combinatorics, no. 12 (December 31, 2017): 1–16. https://doi.org/10.61091/ojac-1211.

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Let \( A \) and \( B \) be additive sets of \( \mathbb{Z}_{2k} \), where \( A \) has cardinality \( k \) and \( B = v \cdot C A \) with \( v \in \mathbb{Z}_{2k}^\times \). In this note, some bounds for the cardinality of \( A + B \) are obtained using four different approaches. We also prove that in a special case, the bound is not sharp and we can recover the whole group as a sumset.

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Amjadi, Jafar, Rana Khoeilar, N. Dehgardi, Lutz Volkmann, and S. M. Sheikholeslami. "The restrained rainbow bondage number of a graph." Tamkang Journal of Mathematics 49, no. 2 (2018): 115–27. http://dx.doi.org/10.5556/j.tkjm.49.2018.2365.

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A restrained $k$-rainbow dominating function (R$k$RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$ such that for any vertex $v \in V (G)$ with $f(v) = \emptyset$ the conditions $\bigcup_{u \in N(v)} f(u)=\{1,2,\ldots,k\}$ and $|N(v)\cap \{u\in V\mid f(u)=\emptyset\}|\ge 1$ are fulfilled, where $N(v)$ is the open neighborhood of $v$. The weight of a restrained $k$-rainbow dominating function is the value $w(f)=\sum_{v\in V}|f (v)|$. The minimum weight of a restrained $k$-rainbow dominating function of $G$ is called the rest

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Ren, Fengyun, Shumin Zhang, and He Li. "Spectral and Sharp Sufficient Conditions for Graphs to Admit a Strong Star Factor." Mathematics 13, no. 10 (2025): 1640. https://doi.org/10.3390/math13101640.

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LetGbe a graph. An odd [1,k]-factor of a graph G is a spanning subgraph H of G such that degH(v) is odd and 1⩽degH(v)⩽k for every v∈V(G) where k is a positive odd integer. We call a spanning subgraph H of a graph G a strong star factor if every component of H is isomorphic to an element of the stars K1,1,K1,2,⋯,K1,r and is an induced subgraph of G where r⩾2 is an integer. In a {K1,1,K1,2,Cm:m⩾3}-factor of G, each component is isomorphic to a member in {K1,1,K1,2,C3,C4⋯,Cm}. A graph G is a strong star factor deleted graph if G−e has a strong star factor for each edge e of G. In this paper, thro

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Chen, Hongyu, and Li Zhang. "A smaller upper bound for the list injective chromatic number of planar graphs." AIMS Mathematics 10, no. 1 (2025): 289–310. https://doi.org/10.3934/math.2025014.

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<p>An injective vertex coloring of a graph $ G $ is a coloring where no two vertices that share a common neighbor are assigned the same color. If for any list $ L $ of permissible colors with size $ k $ assigned to the vertices $ V(G) $ of a graph $ G $, there exists an injective coloring $ \varphi $ in which $ \varphi(v)\in L(v) $ for each vertex $ v\in V(G) $, then $ G $ is said to be injectively $ k $-choosable. The notation $ \chi_{i}^{l}(G) $ represents the minimum value of $ k $ such that a graph $ G $ is injectively $ k $-choosable. In this article, for any maximum degree $ \Delta

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30

Hamja, Jamil, Seyed Mahmoud Sheikholeslami, Imelda S. Aniversario, and Lyster Rey B. Cabardo. "Weakly Connected $k$-Rainbow Domination in Graphs." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5770. https://doi.org/10.29020/nybg.ejpam.v18i2.5770.

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Let $G$ be a simple and connected graph, and let $f$ be a function that assigns to each vertex a set of colors chosen from the set $\{1, 2, 3, \dots, k\}$, i.e., $f : V(G) \to \mathcal P(\{1, 2, 3, \dots, k\})$. If for each vertex $v \in V(G)$ such that $f(v) = \varnothing$, we have $\bigcup_{u \in N_G(v)} f(u) = \{1, 2, 3, \dots, k\},$ then $f$ is called a $k$-rainbow dominating function (kRDF) of $G$. A $k$RDF $f : V(G) \to \mathcal P(\{1,2, \dots, k\})$ is said to be a \textit{weakly connected $k$-rainbow dominating function} (WC$k$RDF) if the set $S = \{v \in V(G) : f(v) \neq \varnothing\}

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31

Cichacz, Sylwia, Agnieszka G�rlich, and Andrea Semani�ov�-Fe�ov��kov�. "Upper bounds on distance vertex irregularity strength of some families of graphs." Opuscula Mathematica 42, no. 4 (2022): 561–71. http://dx.doi.org/10.7494/opmath.2022.42.4.561.

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For a graph \(G\) its distance vertex irregularity strength is the smallest integer \(k\) for which one can find a labeling \(f: V(G)\to \{1, 2, \dots, k\}\) such that \[ \sum_{x\in N(v)}f(x)\neq \sum_{x\in N(u)}f(x)\] for all vertices \(u,v\) of \(G\), where \(N(v)\) is the open neighborhood of \(v\). In this paper we present some upper bounds on distance vertex irregularity strength of general graphs. Moreover, we give upper bounds on distance vertex irregularity strength of hypercubes and trees.

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32

Sun, Yuefang, and Gregory Gutin. "Strong Subgraph Connectivity of Digraphs." Graphs and Combinatorics 37, no. 3 (2021): 951–70. http://dx.doi.org/10.1007/s00373-021-02294-w.

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AbstractLet $$D=(V,A)$$ D = ( V , A ) be a digraph of order n, S a subset of V of size k and $$2\le k\le n$$ 2 ≤ k ≤ n . A strong subgraph H of D is called an S-strong subgraph if $$S\subseteq V(H)$$ S ⊆ V ( H ) . A pair of S-strong subgraphs $$D_1$$ D 1 and $$D_2$$ D 2 are said to be arc-disjoint if $$A(D_1)\cap A(D_2)=\emptyset$$ A ( D 1 ) ∩ A ( D 2 ) = ∅ . A pair of arc-disjoint S-strong subgraphs $$D_1$$ D 1 and $$D_2$$ D 2 are said to be internally disjoint if $$V(D_1)\cap V(D_2)=S$$ V ( D 1 ) ∩ V ( D 2 ) = S . Let $$\kappa _S(D)$$ κ S ( D ) (resp. $$\lambda _S(D)$$ λ S ( D ) ) be the max

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33

Hamja, Jamil, Seyed Mahmoud Sheikholeslami, Aniversario Imelda, and Lyster Rey Cabardo. "Hop $k$-Rainbow Domination in Graphs." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5973. https://doi.org/10.29020/nybg.ejpam.v18i2.5973.

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Let \( G = (V(G), E(G)) \) be a graph. A function \( f \) that assigns to each vertex \linebreak of $G$ a subset of colors from the set \( \{1, 2, \dots, k\} \), i.e., \( f : V(G) \rightarrow P(\{1, 2, 3, \dots, k\}) \), is called a \textit{hop \( k \)-rainbow dominating function} (H$k$RDF) of \( G \) if for every vertex \( v \in V(G)\) with $f(v)= \varnothing$, we have \( \bigcup_{u \in N^{2}_{G}(v)} f(u) = \{1, 2, \dots, k\} \) where $N^{2}_{G}(v)$ is the set of vertices of $G$ at distance two from $v$. The \textit{weight} of \( f \), denoted \( w(f) \), is defined as \( w(f) = \sum_{x \in V

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34

Buyantogtokh, Lkhagva, Batmend Horoldagva, and Kinkar Chandra Das. "On General Reduced Second Zagreb Index of Graphs." Mathematics 10, no. 19 (2022): 3553. http://dx.doi.org/10.3390/math10193553.

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Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant GRMα, known under the name general reduced second Zagreb index, is defined as GRMα(Γ)=∑uv∈E(Γ)(dΓ(u)+α)(dΓ(v)+α), where dΓ(v) is the degree of the vertex v of the graph Γ and α is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs

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35

Wilson, George V. "Mixed Groups of Finite Nilstufe." Canadian Mathematical Bulletin 30, no. 2 (1987): 255–56. http://dx.doi.org/10.4153/cmb-1987-036-7.

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AbstractThis paper constructs a class of examples to show that for torsion-free groups H with finite nilstufe v(H) = n < ∞ there can be divisible torsion groups D with v(H ⊕ D) - n + k for all k ≤ n + 1. This answers a question of Feigelstock. The construction is based on a proposition which bounds v(H ⊕ D) in terms of v(H) and rank (D).

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36

Salindeho, Brilly Maxel, Hilda Assiyatun, and Edy Tri Baskoro. "On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph." Journal of the Indonesian Mathematical Society 26, no. 2 (2020): 175–84. http://dx.doi.org/10.22342/jims.26.2.822.175-184.

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Let c be a k-coloring of a connected graph G and let pi={C1,C2,...,Ck} be the partition of V(G) induced by c. For every vertex v of G, let c_pi(v) be the coordinate of v relative to pi, that is c_pi(v)=(d(v,C1 ),d(v,C2 ),...,d(v,Ck )), where d(v,Ci )=min{d(v,x)|x in Ci }. If every two vertices of G have different coordinates relative to pi, then c is said to be a locating k-coloring of G. The locating-chromatic number of G, denoted by chi_L (G), is the least k such that there exists a locating k-coloring of G. In this paper, we determine the locating-chromatic numbers of some subdivisions of t

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37

Zhou, Sizhong, Yuli Zhang, and Hongxia Liu. "Some properties of (a, b, k)-critical graphs." Filomat 38, no. 16 (2024): 5885–94. https://doi.org/10.2298/fil2416885z.

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Let a, b and k be nonnegative integers with 1 ? a ? b, and let G be a graph with vertex set V(G) and edge set E(G). Then a spanning subgraph F of G is called an [a, b]-factor if a ? dF (v) ? b for any v ? V(G). A graph G is said to be (a, b, k)-critical if G-D contains an [a, b]-factor for each subset D of k elements of V(G). We use |E(G)| and ?(G) to denote the size and spectral radius, respectively. In this paper, we establish a lower bound on the size and spectral radius of a graph G to ensure that G is (a, b, k)-critical, respectively.

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38

Kohlenbach, Ulrich. "Relative constructivity." Journal of Symbolic Logic 63, no. 4 (1998): 1218–38. http://dx.doi.org/10.2307/2586648.

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In a previous paper [13] we introduced a hierarchy (GnAω)n∈ℕ of subsystems of classical arithmetic in all finite types where the growth of definable functions of GnAω corresponds to the well-known Grzegorczyk hierarchy. Let AC-qf denote the schema of quantifier-free choice.[11], [13], [8] and [7] study various analytical principles Γ in the context of the theories GnAω + AC-qf (mainly for n = 2) and use proof-theoretic tools like, e.g., monotone functional interpretation (which was introduced in [12]) to determine their impact on the growth of uniform bounds Φ such thatwhich are extractable fr

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39

Barišić, Ana Klobučar, and Antoaneta Klobučar. "Double total domination number in certain chemical graphs." AIMS Mathematics 7, no. 11 (2022): 19629–40. http://dx.doi.org/10.3934/math.20221076.

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<abstract><p>Let $ G $ be a graph with the vertex set $ V(G) $. A set $ D\subseteq V(G) $ is a total k-dominating set if every vertex $ v\in V(G) $ has at least $ k $ neighbours in $ D $. The total k-domination number $ \gamma_{kt}(G) $ is the cardinality of the smallest total k-dominating set. For $ k = 2 $ the total 2-dominating set is called double total dominating set. In this paper we determine the upper and lower bounds and some exact values for double total domination number on pyrene network $ PY(n) $, $ n\geq 1 $ and hexabenzocoronene $ XC(n) $ $ n\geq 2 $, where pyrene ne

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40

Wang, Hongzhuan, and Piaoyang Yin. "On the eccentricity-based invariants of uniform hypergraphs." Filomat 38, no. 1 (2024): 325–42. http://dx.doi.org/10.2298/fil2401325w.

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Let G =(V, E) be a simple connected hypergraph with V the vertex set and E the edge set, respectively. The eccentricity of vertex v refers to the farthest distance of vertex v from other vertices of G, denoted by ?G(v). The eccentric adjacency index (EAI) of G is described as ?ad(G) = ?u?V(G) SG(u)/?G(u), where SG(u) = ? v?NG(u) dG(v). In this work, we consider the gerneralation of the EAI for hypergraphs to draw several conclusions related to extremal problems to EAI. We first propose several bounds on the EAI of k-uniform hypertrees with fixed maximum degree, diameter and edges, respectively

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41

Lim, Seonhee, Nicolas de Saxcé, and Uri Shapira. "Dimension Bound for Badly Approximable Grids." International Mathematics Research Notices 2019, no. 20 (2018): 6317–46. http://dx.doi.org/10.1093/imrn/rnx330.

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Abstract We show that there exists a subset of full Lebesgue measure $V\subset \mathbb{R}^{n}$ such that for every ϵ > 0 there exists δ > 0 such that for any v ∈ V the dimension of the set of vectors w satisfying $$ \liminf_{k\to\infty} k^{1/n}\langle kv-w\rangle\geqslant \epsilon$$ (where 〈⋅〉 denotes the distance from the nearest integer) is bounded above by n − δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

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42

Gharibyan, Aram H., and Petros A. Petrosyan. "LOCALLY-BALANCED $k$-PARTITIONS OF GRAPHS." Proceedings of the YSU A: Physical and Mathematical Sciences 55, no. 2 (255) (2021): 96–112. http://dx.doi.org/10.46991/pysu:a/2021.55.2.096.

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In this paper we generalize locally-balanced $2$-partitions of graphs and introduce a new notion, the locally-balanced $k$-partitions of graphs, defined as follows: a $k$-partition of a graph $G$ is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$. A $k$-partition ($k\geq 2$) $f$ of a graph $G$ is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A $k$-partition ($k\geq 2$) $f^{\prime}$ of a graph $G$ is a locally-balan

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43

Galatenko, A. V., and V. A. Kuzovikhina. "A Model of Secure Functioning of Computer Systems." Programmnaya Ingeneria 12, no. 3 (2021): 150–56. http://dx.doi.org/10.17587/prin.12.150-156.

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We propose an automata model of computer system security. A system is represented by a finite automaton with states partitioned into two subsets: "secure" and "insecure". System functioning is secure if the number of consecutive insecure states is not greater than some nonnegative integer k. This definition allows one to formally reflect responsiveness to security breaches. The number of all input sequences that preserve security for the given value of k is referred to as a k-secure language. We prove that if a language is k-secure for some natural and automaton V, then it is also k-secure for

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44

Bent-Usman, Wardah Masanggila, Rowena Isla, and Sergio Canoy. "Neighborhood Connected k-Fair Domination Under Some Binary Operations." European Journal of Pure and Applied Mathematics 12, no. 3 (2019): 1337–49. http://dx.doi.org/10.29020/nybg.ejpam.v12i3.3506.

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Let G=(V(G),E(G)) be a simple graph. A neighborhood connected k-fair dominating set (nckfd-set) is a dominating set S subset V(G) such that |N(u) intersection S|=k for every u is an element of V(G)\S and the induced subgraph of S is connected. In this paper, we introduce and invistigate the notion of neighborhood connected k-fair domination in graphs. We also characterize such dominating sets in the join, corona, lexicographic and cartesians products of graphs and determine the exact value or sharp bounds of their corresponding neighborhood connected k-fair domination number.

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45

Käenmäki, Antti, Tuomas Orponen та Laura Venieri. "A Marstrand-type restricted projection theorem in ℝ3". American Journal of Mathematics 147, № 1 (2025): 81–123. https://doi.org/10.1353/ajm.2025.a950274.

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abstract: Marstrand's projection theorem from $1954$ states that if $K \subset \R^{3}$ is an analytic set, then, for $\calH^{2}$ almost every $e \in S^{2}$, the orthogonal projection $\pi_{e}(K)$ of $K$ to the line spanned by $e$ has Hausdorff dimension $\min\{\Hd K,1\}$. This paper contains the following sharper version of Marstrand's theorem. Let $V\subset \R^{3}$ be any $2$-plane, which is not a subspace. Then, for $\calH^{1}$ almost every $e \in S^{2} \cap V$, the projection $\pi_{e}(K)$ has Hausdorff dimension $\min\{\Hd K,1\}$. For $0 \leq t \Hd K$, we also prove an upper bound for the H

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46

Cabrera-Martínez, Abel, Juan Manuel Rueda-Vázquez, and Jaime Segarra. "New Bounds for Three Outer-Independent Domination-Related Parameters in Cactus Graphs." Axioms 13, no. 3 (2024): 177. http://dx.doi.org/10.3390/axioms13030177.

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Let G be a nontrivial connected graph. For a set D⊆V(G), we define D¯=V(G)∖D. The set D is a total outer-independent dominating set of G if |N(v)∩D|≥1 for every vertex v∈V(G) and D¯ is an independent set of G. Moreover, D is a double outer-independent dominating set of G if |N[v]∩D|≥2 for every vertex v∈V(G) and D¯ is an independent set of G. In addition, D is a 2-outer-independent dominating set of G if |N(v)∩D|≥2 for every vertex v∈D¯ and D¯ is an independent set of G. The total, double or 2-outer-independent domination number of G, denoted by γtoi(G), γ×2oi(G) or γ2oi(G), is the minimum car

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47

SUSANTO, FAISAL, KRISTIANA WIJAYA, PRASANTI MIA PURNAMA, and SLAMIN S. "On Distance Irregular Labeling of Disconnected Graphs." Kragujevac Journal of Mathematics 46, no. 4 (2022): 507–23. http://dx.doi.org/10.46793/kgjmat2204.507s.

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A distance irregular k-labeling of a graph G is a function f : V (G) → {1, 2, . . . , k} such that the weights of all vertices are distinct. The weight of a vertex v, denoted by wt(v), is the sum of labels of all vertices adjacent to v (distance 1 from v), that is, wt(v) = P u∈N(v) f(u). If the graph G admits a distance irregular labeling then G is called a distance irregular graph. The distance irregularity strength of G is the minimum k for which G has a distance irregular k-labeling and is denoted by dis(G). In this paper, we derive a new lower bound of distance irregularity strength for gr

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48

GÜRDAL, Verda, and Mualla Birgül HUBAN. "A-Davis-Wielandt-Berezin radius inequalities." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 72, no. 1 (2023): 182–98. http://dx.doi.org/10.31801/cfsuasmas.1107024.

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We consider operator $V$ on the reproducing kernel Hilbert space $\mathcal{H}=\mathcal{H}(\Omega)$ over some set $\Omega$ with the reproducing kernel $K_{\mathcal{H},\lambda}(z)=K(z,\lambda)$ and define A-Davis-Wielandt-Berezin radius $\eta_{A}(V)$ by the formula $\eta_{A}(V):=sup\{\sqrt{| \langle Vk_{\mathcal{H},\lambda},k_{\mathcal{H},\lambda} \rangle_{A}|^{2}+\|Vk_{\mathcal{H},\lambda}\|_{A}^{4}}:\lambda \in \Omega\}$ and $\tilde{V}$ is the Berezin symbol of $V$ where any positive operator $A$-induces a semi-inner product on $\mathcal{H}$ is defined by $\langle x,y \rangle_{A}=\langle Ax,y

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49

Khemmani, Varanoot, and Supachoke Isariyapalakul. "The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes." International Journal of Mathematics and Mathematical Sciences 2018 (August 1, 2018): 1–6. http://dx.doi.org/10.1155/2018/8978193.

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For a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, the multirepresentation of v with respect to W is the k-multiset mr(v∣W)=dv,w1,dv,w2,…,dv,wk, where d(v,wi) is the distance between the vertices v and wi for i=1,2,…,k. The set W is a multiresolving set of G if every two distinct vertices of G have distinct multirepresentations with respect to W. The minimum cardinality of a multiresolving set of G is the multidimension dimM(G) of G. It is shown that, for every pair k,n of integers with k≥3 and n≥3(k-1), there is a connected graph G of order n with dimM(G)=k. For a multi

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50

CZYGRINOW, ANDRZEJ, LOUIS DeBIASIO, H. A. KIERSTEAD, and THEODORE MOLLA. "An Extension of the Hajnal–Szemerédi Theorem to Directed Graphs." Combinatorics, Probability and Computing 24, no. 5 (2014): 754–73. http://dx.doi.org/10.1017/s0963548314000716.

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Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph $\vv G$ with |$\vv G$| = ks and δ($\vv G$) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ($\vv G$)= minv∈V($\vv G$)d−(v)+d+(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. On

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