Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201On the super domination number of graphs83961398010.22049/cco.2019.26587.1122ENJuan AlbertoRodriguez-VelazquezUniversitat Rovira i Virgili0000-0002-9082-7647Douglas 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 $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ 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.http://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, Pakistan0000-0001-8160-4196MubeenJavaidKnowledge Unit of Science
University of Management and Technology, Sialkot 51310, PakistanMarjanMatejicFaculty of Electronic Engineering, 18000 Nis, SerbiaIgorMilovanovicFaculty of Electronic Engineering, Nis, SerbiaEminaMilovanovicFaculty of Electronic Engineering, 18000 Nis, SerbiaJournal Article20190718Let $G=(V,E)$ be a simple connected<br />graph with $n$ vertices, $m$ edges and sequence of vertex degrees<br />$d_1 ge d_2 ge cdots ge d_n>0$, $d_i=d(v_i)$, where $v_iin V$. With $isim j$ we denote adjacency of<br />vertices $v_i$ and $v_j$. The general<br />sum--connectivity index of graph is defined as $chi_{alpha}(G)=sum_{i<br />sim j}(d_i+d_j)^{alpha}$, where $alpha$ is an arbitrary real<br />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.http://comb-opt.azaruniv.ac.ir/article_13987_683b3467bb550bc427ec378858c86a80.pdfAzarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Weak signed Roman domination in graphs1111231398910.22049/cco.2019.26598.1123ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20190625A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as a<br />function $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is the<br />closed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.<br />The weak signed Roman domination number of $G$, denoted by $gamma_{wsR}(G)$, is the minimum weight of a WSRDF in $G$.<br />We initiate the study of the weak signed Roman domination number, and we present different sharp bounds on $gamma_{wsR}(G)$.<br />In addition, we determine the weak signed Roman domination number of some classes of graphs.http://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)0000-0002-0029-2408MAYAMMAJOSEPHCHRIST(Deemed to be University) Hosur Road
Bangalore-5600290000-0001-5819-247XJournal 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 textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. 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.http://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 University0000-0003-3496-277XJournal Article20190625Let $G$ be a finite and simple graph with vertex set $V(G)$. <br />A nonnegative signed total Roman dominating function (NNSTRDF) on a <br />graph $G$ is a function $f:V(G)rightarrow{-1, 1, 2}$ satisfying the conditions<br />that (i) $sum_{xin N(v)}f(x)ge 0$ for each <br />$vin V(G)$, where $N(v)$ is the open neighborhood of $v$, and (ii) every vertex $u$ for which <br />$f(u)=-1$ has a neighbor $v$ for which $f(v)=2$. <br />The weight of an NNSTRDF $f$ is $omega(f)=sum_{vin V (G)}f(v)$. <br />The nonnegative signed total Roman domination number $gamma^{NN}_{stR}(G)$ <br />of $G$ is the minimum weight of an NNSTRDF on $G$. In this paper we<br />initiate the study of the nonnegative signed total Roman domination number <br />of graphs, and we present different bounds on $gamma^{NN}_{stR}(G)$. <br />We determine the nonnegative signed total Roman domination<br />number of some classes of graphs. If $n$ is the order and $m$ the size<br />of the graph $G$, then we show that <br />$gamma^{NN}_{stR}(G)ge frac{3}{4}(sqrt{8n+1}+1)-n$ and $gamma^{NN}_{stR}(G)ge (10n-12m)/5$. <br />In addition, if $G$ is a bipartite graph of order $n$, then we prove<br />that $gamma^{NN}_{stR}(G)ge frac{3}{2}(sqrt{4n+1}-1)-n$.http://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 University0000-0001-9340-4773Journal Article20190409A {em 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 {em 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 $Sigma_{uin V(G)}f(u)$. The {em total Roman domination number} of $G$, $gamma_{tR}(G)$, is the minimum weight of a total Roman dominating function in $G$.<br />The {em total Roman domination subdivision number} ${rm<br />sd}_{gamma_{tR}}(G)$ of a graph $G$ is the minimum number of edges that must be<br />subdivided (each edge in $G$ can be subdivided at most once) in<br />order to increase the total Roman domination number. In this paper,<br />we initiate the study of total Roman domination subdivision<br />number in graphs and we present sharp bounds for this parameter.http://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.2020.26701.1131ENTamasRetiObuda University, Budapest, HungaryAkbarAliUniversity of Hail, Hail, Saudi Arabia0000-0001-8160-4196Journal Article20191125Bell's degree-variance Var$!{}_{B}$ for a graph $G$, with the degree sequence ($d_1,d_2,cdots,d_n$) and size $m$, is defined as<br />$Var!_{B} (G)=frac{1}{n} sum _{i=1}^{n}left[d_{i} -frac{2m}{n}right]^{2}$.<br />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.http://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 Article20190822In this paper we obtain an upper bound and also a lower bound for maximum edges of strongly 2 multiplicative graphs of order n. Also we prove that triangular ladder the graph obtained by duplication of an arbitrary edge by a new vertex in path and the graph<br />obtained by duplicating all vertices by new edges in a path and some other graphs are strongly 2 multiplicativehttp://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 Technology0000-0002-0038-8047RanaKhoeilarAzarbaijan Shahid Madani University0000-0002-2981-3625Seyed MahmoudSheikholeslamiAzarbaijan Shahid Madani University0000-0003-2298-4744Journal Article20200207A signed total double Roman dominating function (STDRDF) on {color{blue}an} isolated-free graph $G=(V,E)$ is a<br />function $f:V(G)rightarrow{-1,1,2,3}$ such that (i) every vertex $v$ with $f(v)=-1$ has at least two<br />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)<br />$sum_{uin N(v)}f(u)ge1$ holds for any vertex $v$.<br />The weight of {color{blue}an} STDRDF is the value $f(V(G))=sum_{uin V(G)}f(u).$ The signed total<br />double Roman domination number $gamma^t_{sdR}(G)$ is the minimum weight of {color{blue}an}<br />STDRDF on $G$. In this paper, we continue the study of the signed total double Roman<br />domination in graphs and present some sharp bounds for this parameter.http://comb-opt.azaruniv.ac.ir/article_14061_e11b84bc918c5f13db5dfa4080bc9852.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 $G=((V(G),E(G))$, a set $Ssubseteq V(G)$ is an outer-weakly convex dominating set if every vertex not in $S$ is adjacent to some vertex in $S$ and $V(G)setminus S$ 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 $G$. An outer-weakly convex dominating set of cardinality $widetilde{gamma}_{wcon}(G)$ will be called a $widetilde{gamma}_{wcon}$-$set$. In this paper, we initiate the study of outer-weakly convex domination as a new variant of graph domination and give some bounds on the outer-weakly convex domination number of a graph. Also, we derived the relations between this parameter to some related domination parameters such as outer-connected domination and outer-convex domination.http://comb-opt.azaruniv.ac.ir/article_14066_7cfa2d7b5265c2cd0a5c1ff4bb021ce5.pdf