Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201On the strength and independence number of powers of paths and cycles7177281472010.22049/cco.2024.29087.1839ENRikioIchishimaDepartment of Sport and Physical Education, Faculty of Physical Education,
Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo 206-8515, JapanFrancisco AntonioMuntaner-BatleGraph Theory and Applications Research Group, School of Electrical Engineering and Computer
Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308, AustraliaYukioTakahashiDepartment of Science and Engineering, Faculty of Electronics and Informatics,
Kokushikan University, 4-28-1 Setagaya, Setagaya-ku, Tokyo 154-8515, JapanJournal Article20231024A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{1,2, \ldots, n \right\}$ to the vertices of $G$. The strength $\mathrm{str}\left(G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right) =\min \left\{ \mathrm{str}_{f}\left( G\right)\left\vert f\text{ is a numbering of }G\right. \right\}$, where $\mathrm{str}_{f}\left( G\right) =\max \left\{ f\left( u\right)+f\left( v\right) \left\vert uv\in E\left( G\right) \right. \right\} $.<br />Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.https://comb-opt.azaruniv.ac.ir/article_14720_9c26bd36c60a54859c209dab3a0cd6ee.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201On connected bipartite $Q$-integral graphs7297421469810.22049/cco.2024.29215.1895ENJesminaPervinDepartment of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University),
Varanasi-221005, IndiaLavanyaSelvaganeshDepartment of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University),
Varanasi-221005, IndiaJournal Article20231122A graph $G$ is said to be $H$-free if $G$ does not contain $H$ as an induced subgraph. Let $\mathcal{S}_{n}^2(m)$ be a \textit{variation of double star $\mathcal{S}_{n}^2$} obtained by adding m (<=n) disjoint edges between the pendant vertices which are at distance 3 in $\mathcal{S}_{n}^2$. A graph having integer eigenvalues for its signless Laplacian matrix is known as a Q-integral graph. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. Any connected Q-integral graph G with Q-spectral radius 7 and maximum edge-degree 8 is either $K_{1,4}\square K_2$ or contains $\mathcal{S}_{4}^2(0)$ as an induced subgraph or is a bipartite graph having at least one of the induced subgraphs $\mathcal{S}_{4}^2(m)$, (m=1, 2, 3). In this article, we improve this result by showing that every connected Q-integral graph G having Q-spectral radius 7, maximum edge-degree 8 is always bipartite and $\mathcal{S}_{4}^2(3)$-free.https://comb-opt.azaruniv.ac.ir/article_14698_0ae966057791b15fb5438f2f90d99bf6.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Leavitt path algebras for order prime Cayley graphs of finite groups7437611469710.22049/cco.2024.28966.1798ENSumantaDasDepartment of Pure Mathematics, University of Calcutta,
35, Ballygunge Circular Road, Kolkata-700019, India0000-0002-8155-2805Mridul KantiSenDepartment of Pure Mathematics, University of Calcutta,
35, Ballygunge Circular Road, Kolkata-700019, IndiaSunil KumarMaityDepartment of Pure Mathematics, University of Calcutta,
35, Ballygunge Circular Road, Kolkata-700019, IndiaJournal Article20230901In this paper, we generalize the concept of Cayley graphs associated to finite groups. The aim of this paper is the characterization of graph theoretic properties of new type of directed graph $\Gamma_P(G;S)$ and algebraic properties of Leavitt path algebra of order prime Cayley graph $O\Gamma(G;S)$, where $G$ is a finite group with a generating set $S$. We show that the Leavitt path algebra of order prime Cayley graph $L_K(O\Gamma(G;S))$ of a non trivial finite group $G$ with any generating set $S$ over a field $K$ is a purely infinite simple ring. Finally, we prove that the Grothendieck group of the Leavitt path algebra $L_K(\Gamma_P(D_n;S))$ is isomorphic to $\mathbb{Z}_{2n-1}$, where $D_n$ is the dihedral group of degree $n$ and $S=\left\{a, b\right\}$ is the generating set of $D_n$.https://comb-opt.azaruniv.ac.ir/article_14697_14ee3edcd14e7fd99b06359abd80d2a7.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Nonlinear inclusion for thermo-electro-elastic: existence, dependence and optimal control7637851471910.22049/cco.2024.29374.1961ENFaizZakariaDepartment of Mathematics, Sultan Moulay Slimane University, FP of Khouribga, MoroccoHichamBenaissaDepartment of Mathematics, Sultan Moulay Slimane University, FP of Khouribga, MoroccoOthmaneBaizIbn Zohr University, Polydisciplinary Faculty of Ouarzazate, MoroccoJournal Article20240116The objective of this paper is to examine a model of a thermo-electro-elastic body situated on a semi-insulator foundation. Friction is characterized by Tresca's friction law, and the contact is bilateral. The primary contribution is to derive the weak variational formulation of the model, constituting a system that couples three inclusions where the unknowns are the strain field, the electric field, and the temperature field. Subsequently, we demonstrate the unique solvability of the system, along with the continuous dependence of its solution under consideration. The secondary contribution involves the investigation of an associated optimal control problem, for which we establish the existence and convergence results. The proofs rely on arguments related to monotonicity, compactness, convex analysis, and lower semicontinuity.https://comb-opt.azaruniv.ac.ir/article_14719_77c7c2fa61fedeacdd8e2843889bd2fb.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201On $e$-super $(a, d)$-edge antimagic total labeling of total graphs of paths and cycles7878021470110.22049/cco.2024.28592.1625ENA.SaibullaDepartment of Mathematics and Actuarial Science,
B.S. Abdur Rahman Crescent Institute of Science and Technology,
Chennai - 600048, Tamil Nadu, IndiaP. Roushini LeelyPushpamDepartment of Mathematic, D.B. Jain College, Chennai - 600097, Tamil Nadu, IndiaJournal Article20230420A $(p, q)$-graph $G$ is $(a, d)$-edge antimagic total if there exists a bijection $f$ from $V(G) \cup E(G)$ to $\{1, 2, \dots, p+q\}$ such that for each edge $uv \in E(G)$, the edge weight $\Lambda(uv) = f(u) + f(uv) + f(v)$ forms an arithmetic progression with first term $a > 0$ and common difference $d \geq 0$. An $(a, d)$-edge antimagic total labeling in which the vertex labels are $1, 2, \dots, p$ and edge labels are $p+1, p+2, \dots, p+q$ is called a {\it super} $(a, d)$-{\it edge antimagic total labeling}. Another variant of $(a, d)$-edge antimagic total labeling called as e-super $(a, d)$-edge antimagic total labeling in which the edge labels are $1, 2, \dots, q$ and vertex labels are $q+1, q+2, \dots, q+p$. In this paper, we investigate the existence of e-super $(a, d)$-edge antimagic total labeling for total graphs of paths, copies of cycles and disjoint union of cycles.https://comb-opt.azaruniv.ac.ir/article_14701_37f7faae4e12bbca5890cfdaa8595a6b.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Total Roman domination and total domination in unit disk graphs8038231469910.22049/cco.2024.28647.1650ENSasmitaRoutDepartment of Mathematics, Indian Institute of Technology Guwahati, Assam, IndiaPawan KumarMishraDepartment of Computer Science and Engineering, IIIT Guwahati, Assam, IndiaGautam KumarDasDepartment of Mathematics, Indian Institute of Technology Guwahati, Assam, IndiaJournal Article20230512Let $G=(V,E)$ be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that each vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u\in V$ with $f(u)=2$. A total Roman dominating function (TRDF) of $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that $(i)$ it is a Roman dominating function, and $(ii)$ the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, $f(V)=\sum_{u\in V} f(u)$, called the total Roman domination number ($\gamma_{tR}(G)$). Similarly, a subset $S\subseteq V$ is said to be a total dominating set (TDS) on the graph $G$ if $(i)$ $S$ is a dominating set of $G$, and $(ii)$ the induced subgraph $G[S]$ does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs. In this paper, we propose a simple $10.5\operatorname{-}$factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is $O(|V|\log k)$, where $k$ is the number of vertices with weights $2$. It is an improvement over the best-known $12$-factor approximation algorithm with running time $O(|V|\log k)$ available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from $8$ to $7.79$. The running time of the proposed algorithm is $O(|V|+|E|)$.https://comb-opt.azaruniv.ac.ir/article_14699_54d35c70e4661b2b1004543b4c5f8160.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Exploring the Precise Edge Irregularity Strength of Generalized Arithmetic and Geometric Staircase Graphs8258351470810.22049/cco.2024.28992.1804ENYeniSusantiDepartment of Mathematics, Universitas Gadjah Mada, IndonesiaMuhammad NurulHudaDepartment of Mathematics, Universitas Gadjah Mada, IndonesiaRamadhani LatiefFirmansyahDepartment of Mathematics, Universitas Gadjah Mada, IndonesiaJournal Article20230912In the context of a finite undirected graph $\zeta$, an edge irregular labelling is defined as a labelling of its vertices with some labels in such a way that each edge has a unique weight, which is determined by the sum of the labels of its endpoints. The main objective of this study is to determine the smallest positive integer $n$ for which it is possible to assign a total edge irregular labelling to $\zeta$ with $n$ as the biggest label. This investigation focuses particularly on cases where $\zeta$ represents the generalized arithmetic and generalized geometric staircase graphs. Within this paper, the definition of generalized geometric staircase graph is proposed. Moreover, we not only establish the edge irregularity strength of these two kind of graphs but also present a method for creating the corresponding edge irregular labelling.https://comb-opt.azaruniv.ac.ir/article_14708_0a0596cd1344b25ae922d6a1a8d383d8.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201A hybrid branch-and-bound and interior-point algorithm for stochastic mixed-integer nonlinear second-order cone programming8378751471410.22049/cco.2024.28898.1768ENHadjerAliouiDepartment of Mathematics, M'Hamed Bougara University of Boumerdés, AlgeriaBahaAlzalgDepartment of Mathematics, The University of Jordan, Amman, JordanJournal Article20230802One of the chief attractions of stochastic mixed-integer second-order cone programming is its diverse applications, especially in engineering (Alzalg and Alioui, {\em IEEE Access}, 10:3522-3547, 2022). The linear and nonlinear versions of this class of optimization problems are still unsolved yet. In this paper, we develop a hybrid optimization algorithm coupling branch-and-bound and primal-dual interior-point methods for solving two-stage stochastic mixed-integer nonlinear second-order cone programming. The adopted approach uses a branch-and-bound technique to handle the integer variables and an infeasible interior-point method to solve continuous relaxations of the resulting subproblems. The proposed hybrid algorithm is also implemented to data to show its efficiency.https://comb-opt.azaruniv.ac.ir/article_14714_d92e5ce711b7c2c8d151822e7400adbd.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-2128104202512012-semi equivelar maps on the torus and the Klein bottle with few vertices8779041470710.22049/cco.2024.29269.1919ENAnand KumarTiwariDepartment of Applied Science, Indian Institute of Information Technology, Allahabad 211 015, IndiaYogendraSinghDepartment of Mathematics and Statistics, Vignan's Foundation for Science, Technology & Research, Vadlamudi 522213, IndiaAmitTripathiDepartment of Applied Science & Humanities, Rajkiya Engineering College, Banda 210201, IndiaJournal Article20231213The $k$-semi equivelar maps, for $k \geq 2$, are generalizations of maps on the surfaces of Johnson solids to closed surfaces other than the 2-sphere. In the present study, we determine 2-semi equivelar maps of curvature 0 exhaustively on the torus and the Klein bottle. Furthermore, we classify (up to isomorphism) all these 2-semi equivelar maps on the surfaces with up to 12 vertices.https://comb-opt.azaruniv.ac.ir/article_14707_11d47d2e2b35663ca3116104f44e6585.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Global Malmquist productivity index for evaluation of multistage series systems with undesirable and non-discretionary data9059311471810.22049/cco.2024.29063.1831ENJafarPourmahmoudDepartment of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranDavoudNorouzi BeneDepartment of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, IranJournal Article20231017Data Envelopment Analysis measures relative efficiency, in which the performances of the DMUs in a group are compared. In this approach, an efficient unit in one group may be considered inefficient compared to the units of other groups and vice versa. To solve this weakness, two known productivity indexes, the Malmquist and Luenberger, have been introduced to evaluate units (or systems) from one period to another. The existence of special types of data such as undesirable and non-discretionary in some multi-stage series systems is unavoidable. The evaluation of such systems in the simultaneous presence of the mentioned data and different periods has not been done so far. Therefore, in this study, we have presented a model with a new approach to evaluate them. At the end of the study, we checked the proposed model’s ability by providing comparative and structural examples. We have shown that without undesirable and non-discretionary data, the proposed is better than other models. Also, this model has been used for the first time and obtained acceptable results in the presence of these data.https://comb-opt.azaruniv.ac.ir/article_14718_506ace4ac3a65a1a60b6680e753f5c3e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201The crossing numbers of join products of $K_4\cup K_1$ with cycles9339481473210.22049/cco.2024.28761.1706ENMichalStašDepartment of Mathematics and Theoretical Informatics,
Faculty of Electrical Engineering and Informatics,
Technical University, 042 00 Košice, Slovak RepublicMariaTimkováDepartment of Mathematics and Theoretical Informatics,
Faculty of Electrical Engineering and Informatics,
Technical University, 042 00 Košice, Slovak RepublicJournal Article20230617The 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, we extend known results concerning crossing numbers of join products of two small graphs with cycles. The crossing number of the join product $G^\ast + C_n$ for the disconnected graph $G^\ast$ consisting of the complete graph $K_{4}$ and one isolated vertex is given, where $C_n$ is the cycle on $n$ vertices. The proof of the main result is done with the help of lemma whose proof is based on a special redrawing technique. Up to now, the crossing numbers of $G + C_n$ were done only for a few disconnected graphs $G$. Finally, by adding new edge to the graph $G^\ast$, we are able to obtain the crossing number of $G_1+C_n$ for one other graph $G_1$ of order five.https://comb-opt.azaruniv.ac.ir/article_14732_3ee57576d44ffa0d3a4f74d7b688f3fe.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Triangular type-2 fuzzy goal programming approach for bimatrix games9499711472610.22049/cco.2024.29061.1827ENJavadTayyebiDepartment of Industrial Engineering, Faculty of Industrial and Computer Engineering,
Birjand University of Technology, Birjand, IranHassanHassanpourDepartment of Mathematics, University of Birjand; Birjand; IranHamidBigdeliInstitute for the Study of War, Army Command and Staff University, Tehran, IranElhamHosseinzadehDepartment of Mathematics; Kosar University of Bojnord, Bojnord; IranJournal Article20231015This paper addresses a bimatrix game in which the satisfactory degrees of the players are vague. Type-2 fuzzy goal programming technique is used to describe the game. Then, the notion of equilibrium points is introduced and an optimization problem is given to calculate them. Moreover, the special case that the type-2 fuzzy goals are triangular is investigated. Finally, an applicable example is presented to illustrate the results.https://comb-opt.azaruniv.ac.ir/article_14726_5208b11ece89ba75d9ae9688d7a815f8.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Complete solutions on local antimagic chromatic number of three families of disconnected graphs9739881472210.22049/cco.2024.29032.1818ENTsz LungChanDepartment of Mathematics,
The Chinese University of Hong Kong,
Shatin, Hong Kong, P.R. ChinaGee-ChoonLauCollege of Computing, Informatics & Mathematics, Universiti Teknologi MARA,
Johor Branch, Segamat Campus, 85000 MalaysiaWai CheeShiuDepartment of Mathematics,
The Chinese University of Hong Kong,
Shatin, Hong Kong, P.R. ChinaJournal Article20231005An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we study local antimagic labeling of disjoint unions of stars, paths and cycles whose components need not be identical. Consequently, we completely determined the local antimagic chromatic numbers of disjoint union of 2 stars, paths, and 2-regular graphs with at most one odd order component respectively.https://comb-opt.azaruniv.ac.ir/article_14722_484086a5c1b730e5028d6b0a64ec958a.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Elliptic Sombor index of chemical graphs9899991475110.22049/cco.2024.29404.1977ENCarlosEspinalInstituto de Matemáticas, Universidad de Antioquia, Medellín, ColombiaIvanGutmanFaculty of Science, University of Kragujevac, 34000 Kragujevac, SerbiaJuanRadaInstituto de Matemáticas, Universidad de Antioquia, Medellín, ColombiaJournal Article20240125Let $G$ be a simple graph. The elliptic Sombor index of $G$ is defined as $$ ESO(G) = \sum_{uv} \left(d_{u}+ d_{v} \right)\sqrt{d^{2}_{u}+d^{2}_{v}},$$ where $d_{u}$ denotes the degree of the vertex $u$, and the sum runs over the set of edges of $G$. In this paper we solve the extremal value problem of $ESO$ over the set of (connected) chemical graphs and over the set of chemical trees, with equal number of vertices. https://comb-opt.azaruniv.ac.ir/article_14751_109c43666449e81cff0582eb50d1713a.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201Set colorings of the Cartesian product of some graph families100110161474810.22049/cco.2024.29090.1840ENMark Anthony C.TolentinoDepartment of Mathematics, Ateneo de Manila University, Quezon City, PhilippinesJanree Ruark C.GatpatanDepartment of Mathematics, Ateneo de Manila University, Quezon City, PhilippinesTimothy Robin Y.TengDepartment of Mathematics, Ateneo de Manila University, Quezon City, PhilippinesJournal Article20231024Neighbor-distinguishing colorings, which are colorings that induce a proper vertex coloring of a graph, have been the focus of different studies in graph theory. One such coloring is the set coloring. For a nontrivial graph $G$, let $c:V(G)\to \mathbb{N}$ and define the neighborhood color set $NC(v)$ of each vertex $v$ as the set containing the colors of all neighbors of $v$. The coloring $c$ is called a set coloring if $NC(u)\neq NC(v)$ for every pair of adjacent vertices $u$ and $v$ of $G$. The minimum number of colors required in a set coloring is called the set chromatic number of $G$ and is denoted by $\chi_s (G)$. In recent years, set colorings have been studied with respect to different graph operations such as join, comb product, middle graph, and total graph. Continuing the theme of these previous works, we aim to investigate set colorings of the Cartesian product of graphs. In this work, we investigate the gap given by $\max\{ \chi_s(G), \chi_s(H) \} - \chi_s(G\ \square\ H)$ for graphs $G$ and $H$. In relation to this objective, we determine the set chromatic numbers of the Cartesian product of some graph families.https://comb-opt.azaruniv.ac.ir/article_14748_ce982efe80e8c0deed488ac2ebd931d3.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810420251201A new measure for transmission irregularity extent of graphs101710371473310.22049/cco.2024.28950.1791ENMahdiehAzariDepartment of Mathematics, Kazerun Branch, Islamic Azad University,
P. O. Box: 73135-168, Kazerun, IranJournal Article20230823The transmission of a vertex ${\varsigma}$ in a connected graph $\mathcal{J}$ is the sum of distances between ${\varsigma}$ and all other vertices of $\mathcal{J}$. A graph $\mathcal{J}$ is called transmission regular if all vertices have the same transmission. In this paper, we propose a new graph invariant for measuring the transmission irregularity extent of transmission irregular graphs. This invariant which we call the total transmission irregularity number (TTI number for short) is defined as the sum of the absolute values of the difference of the vertex transmissions over all unordered vertex pairs of a graph. We investigate some lower and upper bounds on the TTI number which reveal its connection to a number of already established indices. In addition, we compute the TTI number for various families of composite graphs and for some chemical graphs and nanostructures derived from them.https://comb-opt.azaruniv.ac.ir/article_14733_01ac6bbfebb622b598278c6068feadfe.pdf