Communications in Combinatorics and OptimizationCommunications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Sun, 27 Sep 2020 07:06:31 +0100FeedCreatorCommunications in Combinatorics and Optimization
http://comb-opt.azaruniv.ac.ir/
Feed provided by Communications in Combinatorics and Optimization. Click to visit.On the super domination number of graphs
http://comb-opt.azaruniv.ac.ir/article_13980_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100Some new bounds on the general sum--connectivity index
http://comb-opt.azaruniv.ac.ir/article_13987_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100Weak signed Roman domination in graphs
http://comb-opt.azaruniv.ac.ir/article_13989_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100New results on upper domatic number of graphs
http://comb-opt.azaruniv.ac.ir/article_13993_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100Nonnegative signed total Roman domination in graphs
http://comb-opt.azaruniv.ac.ir/article_13992_2294.html
‎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 $gamma^{NN}_{stR}(G)ge frac{3}{2}(sqrt{4n+1}-1)-n$‎.Mon, 30 Nov 2020 20:30:00 +0100Total Roman domination subdivision number in graphs
http://comb-opt.azaruniv.ac.ir/article_13997_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100On the Variance-Type Graph Irregularity Measures
http://comb-opt.azaruniv.ac.ir/article_14026_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100On strongly 2-multiplicative graphs
http://comb-opt.azaruniv.ac.ir/article_14028_2294.html
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 multiplicativeMon, 30 Nov 2020 20:30:00 +0100Weak signed Roman k-domination in graphs
http://comb-opt.azaruniv.ac.ir/article_14019_0.html
Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak signed Roman $k$-domination number $gamma_{wsR}^k(G)$ of $G$ is theminimum weight of a WSRkDF on $G$. In this paper we initiate the study of the weak signed Roman $k$-dominationnumber of graphs, and we present different bounds on $gamma_{wsR}^k(G)$. In addition, we determine theweak signed Roman $k$-domination number of some classes of graphs. Some of our results are extensions ofwell-known properties of the signed Roman $k$-domination number $gamma_{sR}^k(G)$,introduced and investigated by Henning and Volkmann cite{hv16} as well as Ahangar, Henning, Zhao, L"{o}wenstein andSamodivkin cite{ahzls} for the case $k=1$.Thu, 12 Mar 2020 20:30:00 +0100Twin signed total Roman domatic numbers in digraphs
http://comb-opt.azaruniv.ac.ir/article_14024_0.html
Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (ii) every vertex $u$ for which$f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ with$f(v)=f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinct twin signed total Romandominating functions on $D$ with the property that $sum_{i=1}^df_i(v)le 1$for each $vin V(D)$, is called a twin signed total Roman dominating family (offunctions) on $D$. The maximum number of functions in a twin signed total Romandominating family on $D$ is the twin signed total Roman domatic number of $D$,denoted by $d_{stR}^*(D)$. In this paper, we initiate the study of the twinsigned total Roman domatic number in digraphs and we present some sharp bounds on$d_{stR}^*(D)$. In addition, we determine the twin signed total Roman domatic numberof some classes of digraphs.Sat, 14 Mar 2020 20:30:00 +0100Relationships between Randic index and other topological indices
http://comb-opt.azaruniv.ac.ir/article_14043_0.html
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d_u$ denote the degree of vertex $u$ in $G$. The Randi'c index of $G$ is defined as${R}(G) =sum_{uvin E(G)} 1/sqrt{d_ud_v}.$In this paper, we investigate the relationships between Randi'cindex and several topological indices.Sun, 17 May 2020 19:30:00 +0100Bounds on signed total double Roman domination
http://comb-opt.azaruniv.ac.ir/article_14061_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100Outer-weakly convex domination number of graphs
http://comb-opt.azaruniv.ac.ir/article_14066_2294.html
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.Mon, 30 Nov 2020 20:30:00 +0100A note on the first Zagreb index and coindex of graphs
http://comb-opt.azaruniv.ac.ir/article_14047_0.html
Let $G=(V,E)$, $V={v_1,v_2,ldots,v_n}$, be a simple graph with$n$ vertices, $m$ edges and a sequence of vertex degrees$Delta=d_1ge d_2ge cdots ge d_n=delta$, $d_i=d(v_i)$. Ifvertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $isim j$, otherwise, we write $insim j$. The first Zagreb index isvertex-degree-based graph invariant defined as$M_1(G)=sum_{i=1}^nd_i^2$, whereas the first Zagreb coindex isdefined as $overline{M}_1(G)=sum_{insim j}(d_i+d_j)$. A couple of new upper and lower bounds for $M_1(G)$, as well as a new upper boundfor $overline{M}_1(G)$, are obtained.Fri, 22 May 2020 19:30:00 +0100A Generalized form of the Hermite-Hadamard-Fejer type inequalities involving fractional ...
http://comb-opt.azaruniv.ac.ir/article_14060_0.html
Recently, a general class of the Hermit--Hadamard-Fejer inequality on convex functions is studied in [H. Budak, March 2019, 74:29, textit{Results in Mathematics}]. In this paper, we establish a generalization of Hermit--Hadamard--Fejer inequality for fractional integral based on co-ordinated convex functions.Our results generalize and improve several inequalities obtained in earlier studies.Mon, 29 Jun 2020 19:30:00 +0100Independent domination in directed graphs
http://comb-opt.azaruniv.ac.ir/article_14099_0.html
In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, trees, DAGs, cycles, and bipartite graphs. We also provide the idomatic number for special cases of some of these families of digraphs.Sun, 13 Sep 2020 19:30:00 +0100A note on polyomino chains with extremum general sum-connectivity index
http://comb-opt.azaruniv.ac.ir/article_14100_0.html
The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino chain graphs is solved for $alphaFri, 18 Sep 2020 19:30:00 +0100