Communications in Combinatorics and Optimization
Volume:5 Issue: 2, Summer-Autumn 2020

The 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.the formula is not displayed correctly!

Let $G=(V,E)$ be a simple connectedgraph 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 $v_iin V$. With $isim j$ we denote adjacency ofvertices $v_i$ and $v_j$. The generalsum--connectivity index of graph is defined as $chi_{alpha}(G)=sum_{isim j}(d_i+d_j)^{alpha}$, where $alpha$ is an arbitrary realnumber. 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.the formula is not displayed correctly!

A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed 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.

For 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.the formula is not displayed correctly!

‎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 conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎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 $omega(f)=sum_{vin V (G)}f(v)$‎. ‎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 we‎‎initiate 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 domination‎‎number of some classes of graphs‎. ‎If $n$ is the order and $m$ the size‎‎of 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 prove‎‎that $gamma^{NN}_{stR}(G)ge frac{3}{2}(sqrt{4n+1}-1)-n$‎.the formula is not displayed correctly!

A {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 $G$.The {em total Roman domination subdivision number} ${rmsd}_{gamma_{tR}}(G)$ of a graph $G$ is the minimum number of edges that must besubdivided (each edge in $G$ can be subdivided at most once) inorder to increase the total Roman domination number. In this paper,we initiate the study of total Roman domination subdivisionnumber in graphs and we present sharp bounds for this parameter.

Bell'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$Var!_{B} (G)=frac{1}{n} sum _{i=1}^{n}left[d_{i} -frac{2m}{n}right]^{2}$.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.the formula is not displayed correctly!

In 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 graphobtained by duplicating all vertices by new edges in a path and some other graphs are strongly 2 multiplicative

A signed total double Roman dominating function (STDRDF) on {color{blue}an} isolated-free graph $G=(V,E)$ is afunction $f:V(G)rightarrow{-1,1,2,3}$ such that (i) every vertex $v$ with $f(v)=-1$ has at least twoneighbors 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)$sum_{uin N(v)}f(u)ge1$ holds for any vertex $v$.The weight of {color{blue}an} STDRDF is the value $f(V(G))=sum_{uin V(G)}f(u).$ The signed totaldouble Roman domination number $gamma^t_{sdR}(G)$ is the minimum weight of {color{blue}an}STDRDF on $G$. In this paper, we continue the study of the signed total double Romandomination in graphs and present some sharp bounds for this parameter.the formula is not displayed correctly!

For 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.