Transactions on CombinatoricsTransactions on Combinatorics
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Feed provided by Transactions on Combinatorics. Click to visit.On annihilator graph of a finite commutative ring
http://toc.ui.ac.ir/article_20360_3875.html
‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎.Tue, 28 Feb 2017 20:30:00 +0100A neighborhood union condition for fractional $(k,n',m)$-critical deleted graphs
http://toc.ui.ac.ir/article_20355_3875.html
A graph $G$ is called a fractional‎ ‎$(k,n',m)$-critical deleted graph if any $n'$ vertices are removed‎ ‎from $G$ the resulting graph is a fractional $(k,m)$-deleted‎ ‎graph‎. ‎In this paper‎, ‎we prove that for integers $kge 2$‎, ‎$n',mge0$‎, ‎$nge8k+n'+4m-7$‎, ‎and $delta(G)ge k+n'+m$‎, ‎if‎ ‎$$|N_{G}(x)cup N_{G}(y)|gefrac{n+n'}{2}$$‎ ‎for each pair of non-adjacent vertices $x$‎, ‎$y$ of $G$‎, ‎then $G$‎ ‎is a fractional $(k,n',m)$-critical deleted graph‎. ‎The bounds for‎ ‎neighborhood union condition‎, ‎the order $n$ and the minimum degree‎ ‎$delta(G)$ of $G$ are all sharp‎.Tue, 28 Feb 2017 20:30:00 +0100The condition for a sequence to be potentially $A_{L, M}$- graphic
http://toc.ui.ac.ir/article_20361_3875.html
The set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $NS_n$‎. ‎A sequence $piin NS_{n}$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices‎, ‎and such a graph $G$ is called a realization of $pi$‎. ‎The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$‎. ‎The complete product split graph on $L‎ + ‎M$ vertices is denoted by $overline{S}_{L‎, ‎M}=K_{L} vee overline{K}_{M}$‎, ‎where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers‎. ‎Another split graph is denoted by $S_{L‎, ‎M} = overline{S}_{r_{1}‎, ‎s_{1}} veeoverline{S}_{r_{2}‎, ‎s_{2}} vee cdots vee overline{S}_{r_{p}‎, ‎s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$‎. ‎A sequence $pi=(d_{1}‎, ‎d_{2},ldots,d_{n})$ is said to be potentially $S_{L‎, ‎M}$-graphic (respectively $overline{S}_{L‎, ‎M}$)-graphic if there is a realization $G$ of $pi$ containing $S_{L‎, ‎M}$ (respectively $overline{S}_{L‎, ‎M}$) as a subgraph‎. ‎If $pi$ has a realization $G$ containing $S_{L‎, ‎M}$ on those vertices having degrees $d_{1}‎, ‎d_{2},ldots,d_{L+M}$‎, ‎then $pi$ is potentially $A_{L‎, ‎M}$-graphic‎. ‎A non-increasing sequence of non-negative integers $pi = (d_{1}‎, ‎d_{2},ldots,d_{n})$ is potentially $A_{L‎, ‎M}$-graphic if and only if it is potentially $S_{L‎, ‎M}$-graphic‎. ‎In this paper‎, ‎we obtain the sufficient condition for a graphic sequence to be potentially $A_{L‎, ‎M}$-graphic and this result is a generalization of that given by J‎. ‎H‎. ‎Yin on split graphs‎.Tue, 28 Feb 2017 20:30:00 +0100Some properties of comaximal ideal graph of a commutative ring
http://toc.ui.ac.ir/article_20429_3875.html
Let $R$ be a commutative ring with identity‎. ‎We use‎ ‎$varphi (R)$ to denote the comaximal ideal graph‎. ‎The vertices‎ ‎of $varphi (R)$ are proper ideals of R which are not contained‎ ‎in the Jacobson radical of $R$‎, ‎and two vertices $I$ and $J$ are‎ ‎adjacent if and only if $I‎ + ‎J = R$‎. ‎In this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎associated to $varphi (R)$‎.Tue, 28 Feb 2017 20:30:00 +0100A family of $t$-regular self-complementary $k$-hypergraphs
http://toc.ui.ac.ir/article_20363_3875.html
We use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs‎, ‎{em Australas‎. ‎J‎. ‎Combin.}‎, ‎{bf 4} (1991) 229--235.), and present a similar method for constructing $t$-subset-regular‎ ‎self-complementary $k$-uniform hypergraphs of order $v$‎. ‎As an‎ ‎application we show the existence of a new family of 2-subset-regular‎ ‎self-complementary 4-uniform hypergraphs with $v=16m+3$‎.Tue, 28 Feb 2017 20:30:00 +0100On the skew spectral moments of graphs
http://toc.ui.ac.ir/article_20737_3875.html
Let $G$ be a simple graph‎, ‎and $G^{sigma}$‎ ‎be an oriented graph of $G$ with the orientation ‎$sigma$ and skew-adjacency matrix $S(G^{sigma})$‎. ‎The $k-$th skew spectral‎ ‎moment of $G^{sigma}$‎, ‎denoted by‎ ‎$T_k(G^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $G^{sigma}$‎. ‎Suppose‎ ‎$G^{sigma_1}_{1}$ and $G^{sigma_2}_{2}$ are two digraphs‎. ‎If there‎ ‎exists an integer $k$‎, ‎$1 leq k leq n-1$‎, ‎such that for each‎ ‎$i$‎, ‎$0 leq i leq k-1$‎, ‎$T_i(G^{sigma_1}_{1}) =‎ ‎T_i(G^{sigma_2}_{2})$ and‎ ‎$T_k(G^{sigma_1}_{1}) <T_k(G^{sigma_ 2}_{2})$‎ ‎then we write‎ ‎$G^{sigma_1}_{1} prec_{T} G^{sigma_2}_{2}$‎.
‎In this paper‎, ‎we determine some of the skew spectral moments of oriented graphs‎. ‎Also we order some oriented unicyclic graphs with respect to skew spectral moment‎.Tue, 28 Feb 2017 20:30:00 +0100On Numerical Semigroups With Embedding Dimension Three
http://toc.ui.ac.ir/article_20736_0.html
Let $fneq1,3$ be a positive integer‎. ‎We prove that there exists a numerical semigroup $S$ with embedding dimension three such that $f$ is the Frobenius number of $S$‎. ‎We also show that‎ ‎the same fact holds for affine semigroups in higher dimensional monoids‎.Tue, 22 Nov 2016 20:30:00 +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 +0100Full edge-friendly index sets of complete bipartite graphs
http://toc.ui.ac.ir/article_20739_0.html
‎Let $G=(V,E)$ be a simple graph‎. ‎An edge labeling $f:Eto {0,1}$ induces a vertex labeling $f^+:VtoZ_2$ defined by $f^+(v)equiv sumlimits_{uvin E} f(uv)pmod{2}$ for each $v in V$‎, ‎where $Z_2={0,1}$ is the additive group of order 2‎. ‎For $iin{0,1}$‎, ‎let‎ ‎$e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$‎. ‎A labeling $f$ is called edge-friendly if‎ ‎$|e_f(1)-e_f(0)|le 1$‎. ‎$I_f(G)=v_f(1)-v_f(0)$ is called the edge-friendly index of $G$ under an edge-friendly labeling $f$‎. ‎The full edge-friendly index set of a graph $G$ is the set of all possible edge-friendly indices of $G$‎. ‎Full edge-friendly index sets of complete bipartite graphs will be determined‎.Tue, 22 Nov 2016 20:30:00 +0100Adjacent vertex distinguishing acyclic edge coloring of the Cartesian product of graphs
http://toc.ui.ac.ir/article_20988_0.html
‎Let $G$ be a graph and $chi^{prime}_{aa}(G)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $G$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎We prove a general bound for $chi^{prime}_{aa}(Gsquare H)$ for any two graphs $G$ and $H$‎. ‎We also determine‎ ‎exact value of this parameter for the Cartesian product of two paths‎, ‎Cartesian product of a path and a cycle‎, ‎Cartesian product of two trees‎, ‎hypercubes‎. ‎We show that $chi^{prime}_{aa}(C_msquare C_n)$ is at most $6$ fo every $mgeq 3$ and $ngeq 3$‎. ‎Moreover in some cases we find the exact value of $chi^{prime}_{aa}(C_msquare C_n)$‎.Fri, 25 Nov 2016 20:30:00 +0100