Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Skew cyclic codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ with derivation: structural properties and computational results4975171468610.22049/cco.2024.28837.1744ENDjokoSuprijantoCombinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, IndonesiaHopein ChristofenTangCombinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, IndonesiaSchool of Mathematics and Statistics, UNSW, Sydney, AustraliaJournal Article20230714In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $\theta$ and a derivation $\Delta_\theta,$ namely codes as modules over a skew polynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is defined using an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ We investigate the structures of a skew polynomial ring $R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $\Delta_{\theta}$-cyclic codes as well as dual $\Delta_{\theta}$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.https://comb-opt.azaruniv.ac.ir/article_14686_f673327efe5befe212c6887475952e94.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901On the zero forcing number of complementary prism graphs5195301467710.22049/cco.2023.27819.1362ENM.R.RakshaDepartment of Mathematics, CHRIST (Deemed to be university), Bengaluru-560029, Karnataka,
IndiaDepartment of Mathematics, RV College of Engineering, Bengaluru-560059, Karnataka, IndiaCharlesDominicDepartment of Mathematics, CHRIST (Deemed to be university), Bengaluru-560029, Karnataka,
IndiaJournal Article20220518The zero forcing number of a graph is the minimum cardinality among all the zero forcing sets of a graph $G$. The aim of this article is to compute the zero forcing number of complementary prism graphs. Some bounds on the zero forcing number of complementary prism graphs are presented. The remainder of this article discusses the following result. Let $G$ and $\overline{G }$ be connected graphs. Then $Z(G\overline{G})\leq n-1$ if and only if there exists two vertices $v_i,v_j \in V(G)$ and $i\neq j$ such that, either $N(v_i) \subseteq N(v_j)$ or $N[v_i] \subseteq N[v_j]$ in $G$.https://comb-opt.azaruniv.ac.ir/article_14677_1c75a1f96d6a29150a42b286e0fbc827.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901A characterization of locating Roman domination edge critical graphs5315371467810.22049/cco.2023.29108.1853ENH.Abdollahzadeh AhangarDepartment of Mathematics, Babol Noshirvani University of Technology,
Shariati Ave., Babol, I.R. IranH.RahbaniDepartment of Mathematics, Babol Noshirvani University of Technology,
Shariati Ave., Babol, I.R. IranM.R.SadeghiDepartment of Mathematics and Computer Science, Amirkabir University of Technology,
Tehran, I.R. IranJournal Article20230930A Roman dominating function (or just \textit{RDF}) on a graph $G =(V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an \textit{RDF} $f$ is the value $f(V)=\sum_{u \in {V}}f(u)$. An \textit{RDF} $f$ can be represented as $f=(V_0,V_1,V_2)$, where $V_i=\{v\in V:f(v)=i\}$ for $i=0,1,2$. An \textit{RDF} $f=(V_0,V_1,V_2)$ is called a locating Roman dominating function (or just \textit{L\textit{RDF}}) if $N(u)\cap V_2\neq N(v)\cap V_2$ for any pair $u,v$ of distinct vertices of $V_0$. The locating-Roman domination number $\gamma_R^L(G)$ is the minimum weight of an \textit{L\textit{RDF}} of $G$. A graph $G$ is said to be a locating Roman domination edge critical graph, or just $\gamma_R^L$-edge critical graph, if $\gamma_R^L(G-e)>\gamma_R^L(G)$ for all $e\in E$. The purpose of this paper is to characterize the class of $\gamma_R^L$-edge critical graphs.https://comb-opt.azaruniv.ac.ir/article_14678_d3430644fcd1f91fa1590ba89f1494de.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901A note on the re-defined third Zagreb index of trees5395451468110.22049/cco.2023.28868.1757ENNasrinDehgardiDepartment of Mathematics and Computer Science, Sirjan University of Technology,
Sirjan, I.R. IranJournal Article20230724For a graph $\Gamma$, the re-defined third Zagreb index is defined as $$ReZG_3(\Gamma)=\sum_{ab\in E(\Gamma)}\deg_\Gamma(a) \deg_\Gamma(b)\Big(\deg_\Gamma(a)+\deg_\Gamma(b)\Big),$$<br />where $\deg_\Gamma(a)$ is the degree of vertex $a$. We prove for any tree $T$ with $n$ vertices and maximum degree $\Delta$, $ReZG_3(T)\geq16n+\Delta^3+2\Delta^2-13\Delta-26$ when $\Delta< n-1$ and <br />$ReZG_3(T)=n\Delta^2+n\Delta-\Delta^2-\Delta$ when $\Delta=n-1$. <br />Also we determine the corresponding extremal trees. https://comb-opt.azaruniv.ac.ir/article_14681_8606682b87f3d9a6609e5a1020f839f3.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901On the ordering of the Randić index of unicyclic and bicyclic graphs5475611468010.22049/cco.2023.28665.1661ENVenkatesanMaitreyiDepartment of Mathematics, College of Engineering and Technology, Faculty of Engineering and
Technology, SRM Institute of Science and Technology, Kattankulathur, Chengalpet 603 203, IndiaSureshElumalaiDepartment of Mathematics, College of Engineering and Technology, Faculty of Engineering and
Technology, SRM Institute of Science and Technology, Kattankulathur, Chengalpet 603 203, IndiaBala ChandranSelvrajDepartment of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed
University, Thanjavur, IndiaJournal Article20230520Let $d_x$ be the degree of the vertex $x$ in a graph $G$. The Randi<em>ć</em> index of $G$ is defined by $R(G) = \sum_{xy \in E(G)} (d_x d_y)^ {-\frac{1}{2}}$. Recently, Hasni et al. [Unicyclic graphs with Maximum Randi\'{c} indices, Communication in Combinatorics and Optimization, 1 (2023), 161--172] obtained the ninth to thirteenth maximum Randi<em>ć</em> indices among the unicyclic graphs with $n$ vertices. In this paper, we correct the ordering of Randi<em>ć</em> index of unicyclic graphs. In addition, we present the ordering of maximum Randi\'c index among bicyclic graphs of order $n$.https://comb-opt.azaruniv.ac.ir/article_14680_066b31e26a9da5959889a733d012569a.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Intuitionistic fuzzy Sombor indices: A novel approach for improving the performance of vaccination centers5635931467610.22049/cco.2023.28767.1709ENMuhammad KamranJamilDepartment of Mathematics, Riphah International University, Lahore, PakistanShabanAnwerDepartment of Mathematics, Riphah International University, Lahore, PakistanMuhammadAzeemDepartment of Mathematics, Riphah International University, Lahore, PakistanIvanGutmanFaculty of Science, University of Kragujevac, Kragujevac, SerbiaJournal Article20230619Intuitionistic fuzzy graphs are generalizations of fuzzy graphs, in which each vertex is assigned an ordered pair whose first coordinate gives the membership value and the second coordinate gives the non-membership value. There are many theoretical parameters to study different types of graphs and fuzzy graphs, topological indices are one of them. Sombor indices are important in explaining the topology of a graph, and were found to possess useful applicative properties. The two versions of the Sombor indices ($SO_3$ and $SO_4$)are converted into an intuitionistic fuzzy framework, and then formulas for different kinds of graphs are calculated. Our study also involves setting up a network of vaccination centers during a pandemic and applying intuitionistic fuzzy-based topological indices to assess their performance. With the help of this application, we highlight the practical implication and benefits of employing intuitionistic fuzzy-based techniques in vaccination centers. Through a comparative analysis, we evaluate which index is more efficient.https://comb-opt.azaruniv.ac.ir/article_14676_0090fa94ece03092563b5a72af93cb61.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901A new construction of regular and quasi-regular self-complementary graphs5955991468410.22049/cco.2024.28939.1790ENLataKambleDepartment of Mathematics, M.E.S’s Abasaheb Garware College, Pune-411004, IndiaCharusheelaDeshpandeDepartment of Mathematics, College of Engineering Pune, Pune-411006, IndiaBhagyashreeAthawaleDepartment of Mathematics, College of Engineering Pune, Pune-411006, IndiaJournal Article20230822A graph $G$ with a vertex set $V$ and an edge set $E$ is called regular if the degree of every vertex is the same. A quasi-regular graph is a graph whose vertices have one of two degrees $r$ and $r-1$, for some positive integer $r$. A graph $G$ is said to be self-complementary if $G$ is isomorphic to it's complement $\overline{G}$. In this paper we give a new method for construction of regular and quasi-regular self-complementary graph.https://comb-opt.azaruniv.ac.ir/article_14684_7bc19e381aa5f0506f6bdb8d112115ef.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Total coalitions of cubic graphs of order at most 106016151469410.22049/cco.2024.29015.1813ENHamidrezaGolmohammadiNovosibirsk State University, Pirogova str. 2, Novosibirsk, 630090, RussiaSobolev Institute of Mathematics, Ak. Koptyug av. 4, Novosibirsk, 630090, RussiaJournal Article20230927A total coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a total dominating set but whose union $V_{1}\cup V_{2}$, is a total dominating set. A total coalition partition in a <br />graph $G$ of order $n=|V|$ is a vertex partition $\tau = \{V_1, V_2, \dots , V_k \}$ such that every set $V_i \in \tau$ is not a total dominating set but forms a total coalition with another set $V_j\in \tau$ which is not a total dominating set. <br />The total coalition number $TC(G)$ equals the maximum $k$ of a total coalition partition of $G$. In this paper, we determine the total coalition number of all cubic graphs of order $n \le 10$.https://comb-opt.azaruniv.ac.ir/article_14694_a73888ba0e9bf39ceacc7cf7f6c9019a.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Restrained double Roman domatic number6176251468810.22049/cco.2024.28703.1678ENLutzVolkmannLehrstuhl II fur Mathematik, RWTH Aachen University, 52056 Aachen, GermanyJournal Article20230601Let $G$ be a graph with vertex set $V(G)$. A double Roman dominating function (DRDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ mus have at least one neighbor $u$ with $f(u)\ge 2$. If $f$ is a DRDF on $G$, then let $V_0=\{v\in V(G): f(v)=0\}$. A restrained double Roman dominating function is a DRDF $f$ having the property that the subgraph induced by $V_0$ does not have an isolated vertex. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct restrained double Roman dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called a restrained double Roman dominating family (of functions) on $G$. The maximum number of functions in a restrained double Roman dominating family on $G$ is the restrained double Roman domatic number of $G$, denoted by $d_{rdR}(G)$. We initiate the study of the restrained double Roman domatic number, and we present different sharp bounds on $d_{rdR}(G)$. In addition, we determine this parameter for some classes of graphs.https://comb-opt.azaruniv.ac.ir/article_14688_b2ee572088ae6dee4922058c1c7d7b2b.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901A short note on double Roman domination in graphs6276291468510.22049/cco.2024.28751.1704ENAbdelhakOmarLAMDA-RO Laboratory, Department of Mathematics, University of Blida 1, AlgeriaAhmedBouchouDepartment of Mathematics and Computer Science, University of Médéa, Algeria.Journal Article20230614In this short note, we report an erroneous result of Mojdeh, Parsian and Masoumi relating the double Roman domination number to the enclaveless number and the differential of a graph. https://comb-opt.azaruniv.ac.ir/article_14685_2b803213bb8bb2efcd579dface357602.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901On distance Laplacian spectral invariants of brooms and their complements6316561468710.22049/cco.2024.28835.1743ENBilal A.RatherMathematical Science Department, United Arab Emirates University, UAEHilal A.GanieDepartment of School Education, JK Govt. Kashmir, IndiaMustaphaAouchicheMathematical Science Department, United Arab Emirates University, UAEJournal Article20230713For a connected graph $G$ of order $n$, the distance Laplacian matrix $D^L(G)$ is defined as $D^L(G)=Tr(G)-D(G)$, where $Tr(G)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The largest eigenvalue of $D^L(G)$ is the distance Laplacian spectral radius of $G$ and the quantity $DLE(G)=\sum\limits_{i=1}^{n}|\rho^L_i(G)-\frac{2W(G)}{n}|$, where $W(G)$ is the Wiener index of $G$, is the distance Laplacian energy of $G$. Brooms of diameter $4$ are the trees obtained from the path $P_{5}$ by appending pendent vertices at some vertex of $ P_{5}$. One of the interesting and important problems in spectral graph theory is to find extremal graphs for a spectral graph invariant and ordering them according to this graph invariant. This problem has been considered for many families of graphs with respect to different graph matrices. In the present article, we consider this problem for brooms of diameter $4$ and their complements with respect to their distance Laplacian matrix. Formally, we discuss the distance Laplacian spectrum and the distance Laplacian energy of brooms of diameter $4$. We will prove that these families of trees can be ordered in terms of their distance Laplacian energy and the distance Laplacian spectral radius. Further, we obtain the distance Laplacian spectrum and the distance Laplacian energy of complement of the family of double brooms and order them in terms of the smallest non-zero distance Laplacian eigenvalue and the distance Laplacian energy.https://comb-opt.azaruniv.ac.ir/article_14687_6801590615f6f10b9679ffd41c2a0b8b.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Some results on the complete sigraphs with exactly three non-negative eigenvalues6576631468910.22049/cco.2024.29118.1847ENAbbasKermanianDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, IranFaridehHeydariDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran0000-0002-8944-4729MohammadMaghasediDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, IranJournal Article20231030Let $(K_{n},H^-)$ be a complete sigraph of order $n$ whose negative edges induce a subgraph $H$. In this paper, we characterize $(K_n,H^-)$ with exactly 3 non-negative eigenvalues, where $H$ is a non-spanning two-cyclic subgraph of $K_n$.<br /><br />https://comb-opt.azaruniv.ac.ir/article_14689_1e179654c89608c9105b129588b58e45.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901A modified public key cryptography based on generalized Lucas matrices6656791469010.22049/cco.2024.28022.1419ENKalikaPrasadDepartment of Mathematics, Central University of Jharkhand, IndiaDepartment of Mathematics, Government Engineering College, Bhojpur, Bihar, India0000-0002-3653-5854MuneshKumariDepartment of Mathematics, Central University of Jharkhand, IndiaDepartment of Mathematics, Government Engineering College, Bhojpur, Bihar, IndiaHrishikeshMahatoDepartment of Mathematics, Central University of Jharkhand, IndiaJournal Article20221002In this paper, we propose a generalized Lucas matrix (a recursive matrix of higher order) obtained from the generalized Fibonacci sequences. We obtain their algebraic properties such as direct inverse calculation, recursive nature, etc. Then, we propose a modified public key cryptography using the generalized Lucas matrices as a key element that optimizes the keyspace construction complexity. Furthermore, we establish a key agreement for encryption-decryption with a combination of the terms of generalized Lucas sequences under the residue operation.https://comb-opt.azaruniv.ac.ir/article_14690_db630b6d6dcf82443a51ff48559dc659.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Strong $k$-transitive oriented graphs with large minimum degree6816931470610.22049/cco.2024.29025.1815ENMoussaDaamouchKALMA Laboratory, Department of Mathematics, Faculty of Sciences I, Lebanese University, Beirut, LebanonJournal Article20231001A digraph $D=(V,E)$ is $k$-transitive if for any directed $uv$-path of length $k$, we have $(u,v) \in E$. In this paper, we study the structure of strong $k$-transitive oriented graphs having large minimum in- or out-degree. We show that such oriented graphs are \emph{extended cycles}. As a consequence, we prove that Seymour's Second Neighborhood Conjecture (SSNC) holds for $k$-transitive oriented graphs for $k \leq 11$. Also we confirm Bermond--Thomassen Conjecture for $k$-transitive oriented graphs for $k \leq 11$. A characterization of $k$-transitive oriented graphs having a hamiltonian cycle for $k \leq 6$ is obtained immediately.https://comb-opt.azaruniv.ac.ir/article_14706_4b6b9dae3021ff7645c21ca256d07d0f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901Erratum to the paper ``A study on graph topology'' (Published in Commun. Comb. Optim. 8 (2023), no. 2, 397-409.)6957001469510.22049/cco.2024.29290.1932ENPynshngainDharDepartment of Mathematics, North Eastern Hill University, Umshing Mawkynroh,
Shillong 793022, IndiaJohn PaulJala KharbhihDepartment of Mathematics, North Eastern Hill University, Umshing Mawkynroh,
Shillong 793022, IndiaJournal Article20231225In this paper, we will point out errors in Theorem 2, Theorem 4, Theorem 5, Proposition 2, Proposition 3, Theorem 8, and Theorem 9 by giving suitable counterexamples. The statements of Theorem 2, Theorem 5, Proposition 2 and Proposition 3 of this paper have been reformulated and proofs are given.https://comb-opt.azaruniv.ac.ir/article_14695_1bc10eab5556be2d2bda36acfc652f4d.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-212810320250901On co-maximal subgroup graph of $D_n$7017151469610.22049/cco.2024.28396.1528ENAngsumanDasDepartment of Mathematics, Presidency University, Kolkata, IndiaManideepaSahaSchool of Mathematical Sciences, NISER, Bhubaneshwar, IndiaJournal Article20230316Let $G$ be a group and $S$ be the collection of all non-trivial proper subgroups of $G$. The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is defined to be a graph with $S$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. In this paper, we study the comaximal subgroup graph on finite dihedral groups. In particular, we study order, maximum and minimum degree, diameter, girth, domination number, chromatic number and perfectness of comaximal subgroup graph of dihedral groups. Moreover, we prove some isomorphism results on comaximal subgroup graph of dihedral groups.https://comb-opt.azaruniv.ac.ir/article_14696_4df1a1ad9b69aa41f8d0bd3b97a5d62e.pdf