Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201On the super domination number of graphs83961398010.22049/cco.2019.26587.1122ENJuan AlbertoRodríguez-VelázquezUniversitat Rovira i VirgiliDouglas F.KleinTexas A&M UniversityEunjeongYiTexas A&M UniversityJournal Article20190620The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set of $G$ if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets of $G$. In this paper, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. We also obtain results on the super domination number of corona product graphs and Cartesian product graphs.https://comb-opt.azaruniv.ac.ir/article_13980_027a87bda526f67f2d8f3430aa9c2c45.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Some new bounds on the general sum--connectivity index971091398710.22049/cco.2019.26618.1125ENAkbarAliKnowledge Unit of Science
University of Management and Technology, Sialkot 51310, PakistanMubeenJavaidKnowledge Unit of Science
University of Management and Technology, Sialkot 51310, PakistanMarjanMatejićFaculty of Electronic Engineering, 18000 Nis, SerbiaIgorMilovanovićFaculty of Electronic Engineering, Nis, SerbiaEminaMilovanovićFaculty of Electronic Engineering, 18000 Nis, SerbiaJournal Article20190718Let $G=(V,E)$ be a simple connected graph with $n$ vertices, $m$ edges and sequence of vertex degrees $d_1 \ge d_2 \ge \cdots \ge d_n>0$, $d_i=d(v_i)$, where $v_i\in V$. With $i\sim j$ we denote adjacency of vertices $v_i$ and $v_j$. The general sum--connectivity index of graph is defined as $\chi_{\alpha}(G)=\sum_{i\sim j}(d_i+d_j)^{\alpha}$, where $\alpha$ is an arbitrary real number. In this paper we determine relations between $\chi_{\alpha+\beta}(G)$ and $\chi_{\alpha+\beta-1}(G)$, where $\alpha$ and $\beta$ are arbitrary real numbers, and obtain new bounds for $\chi_{\alpha}(G)$. Also, by the appropriate choice of parameters $\alpha$ and $\beta$, we obtain a number of old/new inequalities for different vertex--degree--based topological indices.https://comb-opt.azaruniv.ac.ir/article_13987_cdec3088e115acb1295b55b1ba267a6e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Weak signed Roman domination in graphs1111231398910.22049/cco.2019.26598.1123ENLutzVolkmannRWTH Aachen UniversityJournal Article20190625A weak signed Roman dominating function (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as a function $f:V(G)\rightarrow\{-1,1,2\}$ having the property that $\sum_{x\in N[v]}f(x)\ge 1$ for each $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices. The weak signed Roman domination number of $G$, denoted by $\gamma_{wsR}(G)$, is the minimum weight of a WSRDF in $G$. We initiate the study of the weak signed Roman domination number, and we present different sharp bounds on $\gamma_{wsR}(G)$. In addition, we determine the weak signed Roman domination number of some classes of graphs.https://comb-opt.azaruniv.ac.ir/article_13989_25818cd686d936e0f57852d8ce1b4284.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201New results on upper domatic number of graphs1251371399310.22049/cco.2019.26719.1136ENLibinSamuelCHRIST (Deemed to be University)MAYAMMAJOSEPHCHRIST(Deemed to be University) Hosur Road
Bangalore-560029Journal Article20191210For a graph $G = (V, E)$, a partition $\pi = \{V_1,$ $V_2,$ $\ldots,$ $V_k\}$ of the vertex set $V$ is an \textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j \in \pi$, whenever $i \neq j$. The upper domatic number $D(G)$ is the maximum order of an upper domatic partition of $G$. We study the properties of upper domatic number and propose an upper bound in terms of clique number. Further, we discuss the upper domatic number of certain graph classes including unicyclic graphs and power graphs of paths and cycles.https://comb-opt.azaruniv.ac.ir/article_13993_d2d2bdfc3ac890ae53ac04a1d2ad425e.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Nonnegative signed total Roman domination in graphs1391551399210.22049/cco.2019.26599.1124ENNasrinDehgardiSirjan University of Technology, Sirjan 78137, IranLutzVolkmannRWTH Aachen UniversityJournal Article20190625<br />Let $G$ be a finite and simple graph with vertex set $V(G)$. A nonnegative signed total Roman dominating function (NNSTRDF) on a graph $G$ is a function $f:V(G)\rightarrow\{-1, 1, 2\}$ satisfying the conditions that (i) $\sum_{x\in N(v)}f(x)\ge 0$ for each $v\in V(G)$, where $N(v)$ is the open neighborhood of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has a neighbor $v$ for which $f(v)=2$. The weight of an NNSTRDF $f$ is $\omega(f)=\sum_{v\in V (G)}f(v)$. The nonnegative signed total Roman domination number $\gamma^{NN}_{stR}(G)$ of $G$ is the minimum weight of an NNSTRDF on $G$. In this paper we initiate the study of the nonnegative signed total Roman domination number of graphs, and we present different bounds on $\gamma^{NN}_{stR}(G)$. We determine the nonnegative signed total Roman domination number of some classes of graphs. If $n$ is the order and $m$ is the size of the graph $G$, then we show that $\gamma^{NN}_{stR}(G)\ge \frac{3}{4}(\sqrt{8n+1}+1)-n$ and $\gamma^{NN}_{stR}(G)\ge (10n-12m)/5$. In addition, if $G$ is a bipartite graph of order $n$, then we prove that $\gamma^{NN}_{stR}(G)\ge \frac{3}{2}\sqrt{4n+1}-1)-n$.https://comb-opt.azaruniv.ac.ir/article_13992_95a6741eac9ee78064b669bd3e8a9b20.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Total Roman domination subdivision number in graphs1571681399710.22049/cco.2020.26470.1117ENJafarAmjadiAzarbaijan Shahid Madani UniversityJournal Article20190409A Roman dominating function on a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A total Roman dominating function is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertices. The weight of a total Roman dominating function $f$ is the value $\Sigma_{u\in V(G)}f(u)$. The total Roman domination number of $G$, $\gamma_{tR}(G)$, is the minimum weight of a total Roman dominating function on $G$. The total Roman domination subdivision number ${\rm sd}_{\gamma_{tR}}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the total Roman domination number. In this paper, we initiate the study of total Roman domination subdivision number in graphs and we present sharp bounds for this parameter. <br /><br />https://comb-opt.azaruniv.ac.ir/article_13997_361f34d42a9e4c4eeb3798140a18ae6f.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201On the Variance-Type Graph Irregularity Measures1691781402610.22049/cco.2020.26701.1131ENTamasRétiObuda University, Budapest, HungaryAkbarAliUniversity of Hail, Hail, Saudi ArabiaJournal Article20191125Bell's degree-variance Var$\!{}_{B}$ for a graph $G$, with the degree sequence ($d_1,d_2,\ldots,d_n$) and size $m$, is defined as $Var\!_{B} (G)=\frac{1}{n} \sum _{i=1}^{n}\left[d_{i} -\frac{2m}{n}\right]^{2}$. In this paper, a new version of the irregularity measures of variance-type, denoted by $Var_q$, is introduced and discussed. Based on a comparative study, it is demonstrated that the newly proposed irregularity measure $Var_q$ possess a better discrimination ability than the classical Bell's degree-variance in several cases.https://comb-opt.azaruniv.ac.ir/article_14026_0dc3a405cc9c0e6b0b9e2fa92c09cff8.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201On strongly 2-multiplicative graphs1791901402810.22049/cco.2020.26647.1127END.D.SomashekaraDepartment of Studies in Mathematics, University of Mysore
Manasagangotri, Mysore-570006, IndiaH.E.RaviDepartment of Studies in Mathematics,
University of Mysore, Manasagangotri, Mysore-570006C.R.VeenaDepartment of Mathematics, JSS College of Arts, Commerce and Science,
Mysore-570025, IndiaJournal Article20190822A simple connected graph $G$ of order $n\ge 3$ is a strongly 2-multiplicative if there is an injective mapping $f:V(G)\rightarrow \{1,2,\ldots,n\}$ such that the induced mapping $h:\mathcal{A} \rightarrow \mathbb{Z}^+$ defined by $h(\mathcal{P})= \prod_{i=1}^{3} f({v_j}_i)$, where $j_1,j_2,j_{3}\in \{1,2,\ldots,n\}$, and $\mathcal{P}$ is the path homotopy class of paths having the vertex set $\{ v_{j_1}, v_{j_2},v_{j_{3}} \}$, is injective. Let $\Lambda(n)$ be the number of distinct path homotopy classes in a strongly 2-multiplicative graph of order $n$. In this paper we obtain an upper bound and also a lower bound for $\Lambda(n)$. Also we prove that triangular ladder, $P_{2} \bigodot C_{n}$, $P_{m}\bigodot P_{n}$, the graph obtained by duplication of an arbitrary edge by a new vertex in path $P_{n}$ and the graph obtained by duplicating all vertices by new edges in a path $P_{n}$ are strongly 2-multiplicative. https://comb-opt.azaruniv.ac.ir/article_14028_5ef7f3d3936254933ebe84c316170400.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Bounds on signed total double Roman domination1912061406110.22049/cco.2020.26761.1140ENL.ShahbaziAzarbaijan Shahid Madani UniversityH.Abdollahzadeh AhangarBabol Noshirvani University of TechnologyR.KhoeilarAzarbaijan Shahid Madani UniversitySeyed MahmoudSheikholeslamiAzarbaijan Shahid Madani UniversityJournal Article20200207A signed total double Roman dominating function (STDRDF) on {an} isolated-free graph $G=(V,E)$ is a function $f:V(G)\rightarrow\{-1,1,2,3\}$ such that (i) every vertex $v$ with $f(v)=-1$ has at least two neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, (ii) every vertex $v$ with $f(v)=1$ has at least one neighbor $w$ with $f(w)\geq2$ and (iii) $\sum_{u\in N(v)}f(u)\ge1$ holds for any vertex $v$. The weight of {an} STDRDF is the value $f(V(G))=\sum_{u\in V(G)}f(u).$ The signed total double Roman domination number $\gamma^t_{sdR}(G)$ is the minimum weight of an STDRDF on $G$. In this paper, we continue the study of the signed total double Roman domination in graphs and present some sharp bounds for this parameter.https://comb-opt.azaruniv.ac.ir/article_14061_9ffaa7643dce5a215c51f8746496b9f9.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Outer-weakly convex domination number of graphs2072151406610.22049/cco.2020.26871.1154ENJonecis ADayapUniversity of San Jose-RecoletosRichardAlcantaraUniversity of CebuRomaAnoosCebu Technological University-San Fernando ExtensionJournal Article20200515For a given simple graph $G=(V,E)$, a set $S\subseteq V$ is an outer-weakly convex dominating set if every vertex in $V\setminus S$ is adjacent to some vertex in $S$ and $V\setminus S$ is a weakly convex set. The \emph{outer-weakly convex domination number} of a graph $G$, denoted by $\widetilde{\gamma}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating set of $G$. In this paper, we initiate the study of outer-weakly convex domination as a new variant of graph domination and we show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on $\widetilde{\gamma}_{wcon}(G)$ and, for some particular families of graphs, we obtain closed formula. https://comb-opt.azaruniv.ac.ir/article_14066_322f3d041cb8dc320b968fd5222905f9.pdf