Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.Zero-sum flow number of categorical and strong product of graphs
http://toc.ui.ac.ir/article_24517_4442.html
A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum $k$-flow if the absolute values of edges are less than $k$. We define the zero-sum flow number of $G$ as the least integer $k$ for which $G$ admitting a zero sum $k$-flow.? In this paper we gave complete zero-sum flow and zero sum numbers for categorical and strong product of two graphs namely cycle and paths.Mon, 30 Nov 2020 20:30:00 +0100Further results on maximal rainbow domination number
http://toc.ui.ac.ir/article_24579_4442.html
‎A 2-rainbow dominating function (2RDF) of a graph $G$ is a‎ ‎function $f$ from the vertex set $V(G)$ to the set of all subsets‎ ‎of the set ${1,2}$ such that for any vertex $vin V(G)$ with‎ ‎$f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$‎ ‎is fulfilled‎, ‎where $N(v)$ is the open neighborhood of $v$‎. ‎A ‎ ‎maximal 2-rainbow dominating function of a graph $G$ is a ‎‎$‎‎2‎$‎-rainbow dominating function $f$ such that the set ${win‎‎V(G)|f(w)=emptyset}$ is not a dominating set of $G$‎. ‎The‎ ‎weight of a maximal 2RDF $f$ is the value $omega(f)=sum_{vin‎ ‎V}|f (v)|$‎. ‎The maximal $2$-rainbow domination number of a‎ ‎graph $G$‎, ‎denoted by $gamma_{m2r}(G)$‎, ‎is the minimum weight of a‎ ‎maximal 2RDF of $G$‎. ‎In this paper‎, ‎we continue the study of maximal‎ ‎2-rainbow domination {number} in graphs‎. ‎Specially‎, ‎we first characterize all graphs with large‎ ‎maximal 2-rainbow domination number‎. ‎Finally‎, ‎we determine the maximal ‎$‎2‎$‎‎-‎rainbow domination number in the sun and sunlet graphs‎.Mon, 30 Nov 2020 20:30:00 +0100Edge-group choosability of outerplanar and near-outerplanar graphs
http://toc.ui.ac.ir/article_24806_4442.html
Let $chi_{gl}(G)$ be the {it{group choice number}} of $G$. A graph $G$ is called {it{edge-$k$-group choosable}} if its line graph is $k$-group choosable. The {it{group-choice index}} of $G$, $chi'_{gl}(G)$, is the smallest $k$ such that $G$ is edge-$k$-group choosable, that is, $chi'_{gl}(G)$ is the group chice number of the line graph of $G$, $chi_{gl}(ell(G))$. It is proved that, if $G$ is an outerplanar graph with maximum degree $D<5$, or if $G$ is a $({K_2}^c+(K_1 cup K_2))$-minor-free graph, then $chi'_{gl}(G)leq D(G)+1$. As a straightforward consequence, every $K_{2,3}$-minor-free graph $G$ or every $K_4$-minor-free graph $G$ is edge-$(D(G)+1)$-group choosable. Moreover, it is proved that if $G$ is an outerplanar graph with maximum degree $Dgeq 5$, then $chi'_{gl}(G)leq D$.Mon, 30 Nov 2020 20:30:00 +0100On the dominated chromatic number of certain graphs
http://toc.ui.ac.ir/article_24862_4442.html
‎Let $G$ be a simple graph‎. ‎The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex‎. ‎The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$‎, ‎denoted by $chi_{dom}(G)$‎. ‎Stability (bondage number) of dominated chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the dominated chromatic number of $G$‎. ‎In this paper‎, ‎we study the dominated chromatic number‎, ‎dominated stability and dominated bondage number of certain graphs‎.
‎Mon, 30 Nov 2020 20:30:00 +0100Gutman index, edge-Wiener index and edge-connectivity
http://toc.ui.ac.ir/article_24868_4442.html
‎We study the Gutman index ${rm Gut}(G)$ and the edge-Wiener index $W_e (G)$ of connected graphs $G$ of given order $n$ and edge-connectivity $lambda$‎. ‎We show that the bound ${rm Gut}(G) le frac{2^4 cdot 3}{5^5 (lambda+1)} n^5‎ + ‎O(n^4)$ is asymptotically tight for $lambda ge 8$‎. ‎We improve this result considerably for $lambda le 7$ by presenting asymptotically tight upper bounds on ${rm Gut}(G)$ and $W_e (G)$ for $2 le lambda le 7$‎.Mon, 30 Nov 2020 20:30:00 +0100On a Conjecture about Degree Deviation Measure of Graphs
http://toc.ui.ac.ir/article_24803_0.html
Let $G$ be an $n-$vertex graph with $m$ vertices‎. ‎The degree deviation measure of $G$ is defined as‎ ‎$s(G)$ $=$ $sum_{vin V(G)}|deg_G(v)‎- ‎frac{2m}{n}|,$ where $n$ and $m$ are the number of vertices and edges of $G$‎, ‎respectively‎. ‎The aim of this paper is to prove the Conjecture 4.2 of [J‎. ‎A‎. ‎de Oliveira‎, ‎C‎. ‎S‎. ‎Oliveira‎, ‎C‎. ‎Justel and N‎. ‎M‎. ‎Maia de Abreu‎, ‎Measures of irregularity of graphs‎, ‎emph{Pesq‎. ‎Oper.}, textbf{33} (2013) 383--398]‎. ‎The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed‎.Sun, 19 Jul 2020 19:30:00 +0100Some inequalities involving the distance signless Laplacian eigenvalues of graphs
http://toc.ui.ac.ir/article_24869_0.html
‎Given a simple graph $G$‎, ‎the distance signlesss Laplacian‎ ‎$D^{Q}(G)=Tr(G)+D(G)$ is the sum of vertex transmissions matrix‎ ‎$Tr(G)$ and distance matrix $D(G)$‎. ‎In this paper‎, ‎thanks to the‎ ‎symmetry of $D^{Q}(G)$‎, ‎we obtain novel sharp bounds on the distance‎ ‎signless Laplacian eigenvalues of $G$‎, ‎and in particular the‎ ‎distance signless Laplacian spectral radius‎. ‎The bounds are‎ ‎expressed through graph diameter‎, ‎vertex covering number‎, ‎edge‎ ‎covering number‎, ‎clique number‎, ‎independence number‎, ‎domination‎ ‎number as well as extremal transmission degrees‎. ‎The graphs‎ ‎achieving the corresponding bounds are delineated‎. ‎In addition‎, ‎we‎ ‎investigate the distance signless Laplacian spectrum induced by‎ ‎Indu-Bala product‎, ‎Cartesian product as well as extended double‎ ‎cover graph‎.Sun, 09 Aug 2020 19:30:00 +0100Statistics on restricted Fibonacci words
http://toc.ui.ac.ir/article_24874_0.html
We study two foremost Mahonian statistics, the major index and the inversion number for a class of binary words called restricted Fibonacci words. The language of restricted Fibonacci words satisfies recurrences which allow for the calculation of the generating functions in two different ways. These yield identities involving the $q$-binomial coefficients and provide non-standard $q$-analogues of the Fibonacci numbers. The major index generating function for restricted Fibonacci words turns out to be a $q$-power multiple of the inversion generating function.Thu, 13 Aug 2020 19:30:00 +0100Symmetric $1$-designs from $PSL_{2}(q),$ for $q$ a power of an odd prime
http://toc.ui.ac.ir/article_24875_0.html
Let $G = PSL_{2}(q)$‎, ‎where $q$ is a power of an odd prime‎. ‎Let $M$ be a maximal subgroup of $G$‎. ‎Define $leftlbrace frac{|M|}{|M cap M^g|}‎: ‎g in G rightrbrace$ to be the set of orbit lengths of the primitive action of $G$ on the conjugates of a maximal subgroup $M$ of $G.$ By using a method described by Key and Moori in the literature‎, ‎we construct all primitive symmetric $1$-designs that admit $G$ as a permutation group of automorphisms‎.Thu, 13 Aug 2020 19:30:00 +0100Some remarks on the sum of powers of the degrees of graphs
http://toc.ui.ac.ir/article_24952_0.html
‎Let $G=(V,E)$ be a simple graph with $nge 3$ vertices‎, ‎$m$ edges‎ ‎and vertex degree sequence $Delta=d_1 ge d_2 ge cdots ge‎ ‎d_n=delta>0$‎. ‎Denote by $S={1, 2,ldots,n}$ an index set and by‎ ‎$J={I=(r_1, r_2,ldots,r_k) ‎, ‎| ‎, ‎1le r_1<r_2<cdots<r_kle‎ ‎n}$ a set of all subsets of $S$ of cardinality $k$‎, ‎$1le kle‎ ‎n-1$‎. ‎In addition‎, ‎denote by‎ $d_{I}=d_{r_1}+d_{r_2}+cdots+d_{r_k}$‎, ‎$1le kle n-1$‎, ‎$1le‎ ‎r_1<r_2<cdots<r_kle n-1$‎, ‎the sum of $k$ arbitrary vertex‎ ‎degrees‎, ‎where $Delta_{I}=d_{1}+d_{2}+cdots+d_{k}$ and‎ ‎$delta_{I}=d_{n-k+1}+d_{n-k+2}+cdots+d_{n}$‎. ‎We consider the following graph invariant‎ ‎$S_{alpha,k}(G)=sum_{Iin J}d_I^{alpha}$‎, ‎where $alpha$ is an‎ ‎arbitrary real number‎, ‎and establish its bounds‎. ‎A number of known bounds for various topological indices are obtained as special cases‎.Tue, 15 Sep 2020 19:30:00 +0100