Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.Products of graphs and Nordhaus-Gaddum type inequalities for eigenvalues
http://toc.ui.ac.ir/article_21474_0.html
In this paper‎, ‎we obtain $alpha$ as coefficient for the $G=K_{alpha n} cup overline{K_{(1-alpha)n}}$ and by which we discuss Nikiforov's conjecture for $lambda_{1}$ and Aouchiche and Hansen's conjecture for $q_1$ in Nordhaus-Gaddum type inequalities‎. ‎Furthermore‎, ‎by the properties of the products of graphs we put forward a new approach to find some bounds of Nordhaus-Gaddum type inequalities‎.Thu, 08 Jun 2017 19:30:00 +0100The central vertices and radius of the regular graph of ideals
http://toc.ui.ac.ir/article_21472_3875.html
The regular graph of ideals of the commutative ring $R$‎, ‎denoted by ${Gamma_{reg}}(R)$‎, ‎is a graph whose vertex‎ ‎set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element‎. ‎In this paper‎, ‎it is proved that the radius of $Gamma_{reg}(R)$ equals $3$‎. ‎The central vertices of $Gamma_{reg}(R)$ are determined‎, ‎too‎.Thu, 30 Nov 2017 20:30:00 +0100The harmonic index of subdivision graphs
http://toc.ui.ac.ir/article_21471_3875.html
‎The harmonic index of a graph $G$ is defined as the sum of the weights‎ ‎$frac{2}{deg_G(u)+deg_G(v)}$ of all edges $uv$‎ ‎of $G$‎, ‎where $deg_G(u)$ denotes the degree of a vertex $u$ in $G$‎. ‎In this paper‎, ‎we study the harmonic index of subdivision graphs‎, ‎$t$-subdivision graphs and also‎, ‎$S$-sum and $S_t$-sum of graphs‎.Thu, 30 Nov 2017 20:30:00 +0100Annihilating submodule graph for modules
http://toc.ui.ac.ir/article_21462_0.html
Let $R$ be a commutative ring and $M$ an‎ ‎$R$-module‎. ‎In this article‎, ‎we introduce a new generalization of‎ ‎the annihilating-ideal graph of commutative rings to modules‎. ‎The‎ ‎annihilating submodule graph of $M$‎, ‎denoted by $Bbb G(M)$‎, ‎is an‎ ‎undirected graph with vertex set $Bbb A^*(M)$ and two distinct‎ ‎elements $N$ and $K$ of $Bbb A^*(M)$ are adjacent if $N*K=0$‎. ‎In‎ ‎this paper we show that $Bbb G(M)$ is a connected graph‎, ‎${rm‎ ‎diam}(Bbb G(M))leq 3$‎, ‎and ${rm gr}(Bbb G(M))leq 4$ if $Bbb‎ ‎G(M)$ contains a cycle‎. ‎Moreover‎, ‎$Bbb G(M)$ is an empty graph‎ ‎if and only if ${rm ann}(M)$ is a prime ideal of $R$ and $Bbb‎ ‎A^*(M)neq Bbb S(M)setminus {0}$ if and only if $M$ is a‎ ‎uniform $R$-module‎, ‎${rm ann}(M)$ is a semi-prime ideal of $R$‎ ‎and $Bbb A^*(M)neq Bbb S(M)setminus {0}$‎. ‎Furthermore‎, ‎$R$‎ ‎is a field if and only if $Bbb G(M)$ is a complete graph‎, ‎for‎ ‎every $Min R-{rm Mod}$‎. ‎If $R$ is a domain‎, ‎for every divisible‎ ‎module $Min R-{rm Mod}$‎, ‎$Bbb G(M)$ is a complete graph with‎ ‎$Bbb A^*(M)=Bbb S(M)setminus {0}$‎. ‎Among other things‎, ‎the‎ ‎properties of a reduced $R$-module $M$ are investigated when‎ ‎$Bbb G(M)$ is a bipartite graph‎.Wed, 07 Jun 2017 19:30:00 +0100Some topological indices and graph properties
http://toc.ui.ac.ir/article_21467_0.html
In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected or $k$-connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.Wed, 07 Jun 2017 19:30:00 +0100Majorization and the number of bipartite graphs for given vertex degrees
http://toc.ui.ac.ir/article_21469_0.html
The emph{bipartite realisation problem} asks for a pair of non-negative‎, ‎non-increasing integer lists $a:=(a_1,ldots,a_n)$ and $b:=(b_1,ldots,b_{n'})$ if there is a labeled bipartite graph $G(U,V,E)$ (no loops or multiple edges) such that each vertex $u_i in U$ has degree $a_i$ and each vertex $v_i in V$ degree $b_i.$ The Gale-Ryser theorem provides characterisations for the existence of a `realisation' $G(U,V,E)$ that are strongly related to the concept of emph{majorisation}‎. ‎We prove a generalisation; list pair $(a,b)$ has more realisations than $(a',b),$ if $a'$ majorises $a.$ Furthermore‎, ‎we give explicitly list pairs which possess the largest number of realisations under all $(a,b)$ with fixed $n$‎, ‎$n'$ and $m:=sum_{i=1}^n a_i.$ We introduce the notion~emph{minconvex list pairs} for them‎. ‎If $n$ and $n'$ divide $m,$ minconvex list pairs turn in the special case of two constant lists $a=(frac{m}{n},ldots,frac{m}{n})$ and $b=(frac{m}{n'},ldots,frac{m}{n'}).$‎Wed, 07 Jun 2017 19:30:00 +0100On the average eccentricity, the harmonic index and the largest signless Laplacian ...
http://toc.ui.ac.ir/article_21470_0.html
The eccentricity of a vertex is the maximum distance from it to‎ ‎another vertex and the average eccentricity $eccleft(Gright)$ of a‎ ‎graph $G$ is the mean value of eccentricities of all vertices of‎ ‎$G$‎. ‎The harmonic index $Hleft(Gright)$ of a graph $G$ is defined‎ ‎as the sum of $frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of‎ ‎$G$‎, ‎where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$‎. ‎In‎ ‎this paper‎, ‎we determine the unique tree with minimum average‎ ‎eccentricity among the set of trees with given number of pendent‎ ‎vertices and determine the unique tree with maximum average‎ ‎eccentricity among the set of $n$-vertex trees with two adjacent‎ ‎vertices of maximum degree $Delta$‎, ‎where $ngeq 2Delta$‎. ‎Also‎, ‎we‎ ‎give some relations between the average eccentricity‎, ‎the harmonic‎ ‎index and the largest signless Laplacian eigenvalue‎, ‎and strengthen‎ ‎a result on the Randi'{c} index and the largest signless Laplacian‎ ‎eigenvalue conjectured by Hansen and Lucas cite{hl}‎.Wed, 07 Jun 2017 19:30:00 +0100New class of integral bipartite graphs with large diameter
http://toc.ui.ac.ir/article_20738_0.html
In this paper‎, ‎we construct a new class of integral bipartite graphs (not necessarily trees) with large even diameters‎. ‎In fact‎, ‎for every finite set $A$ of positive integers of size $k$ we construct an integral bipartite graph $G$ of diameter $2k$ such that the set of positive eigenvalues of $G$ is exactly $A$‎. ‎This class of integral bipartite graphs has never found before‎.Tue, 22 Nov 2016 20:30:00 +0100