University of IsfahanTransactions on Combinatorics2251-86577120180301Annihilating submodule graph for modules1122146210.22108/toc.2017.21462ENSaeedSafaeeyanDepartment of mathematical Sciences, Yasouj university,Yasouj, 75918-74831, IRAN.Journal Article20150417Let $R$ 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 $M\in R-{\rm Mod}$. If $R$ is a domain, for every divisible module $M\in R-{\rm Mod}$, $\Bbb G(M)$ 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.https://toc.ui.ac.ir/article_21462_0dc774d0c7d8fb2042f09cc2cf66d2ad.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577120180301New class of integral bipartite graphs with large diameter13172073810.22108/toc.2016.20738ENAlirezaFiuj LaaliShahed university, Tehran, Iran.HamidHaj Seyyed JavadiShahed university, Tehran, Iran.Journal Article20160204In 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.https://toc.ui.ac.ir/article_20738_e02d3bdd82972786f02e60e8bfbe4497.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577120180301Majorization and the number of bipartite graphs for given vertex degrees19302146910.22108/toc.2017.21469ENAnnabellBergerDepartment of Computer Science
Martin-Luther University Halle-WittenbergJournal Article20170131The \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 $b=(\frac{m}{n'},\ldots,\frac{m}{n'}).$https://toc.ui.ac.ir/article_21469_4550ee854fabeca9498c8036d49fbd87.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577120180301Products of graphs and Nordhaus-Gaddum type inequalities for eigenvalues31362147410.22108/toc.2017.21474ENNastaranKeyvanAmirkabir University of TechnologyFarhadRahmatiAmirkabir University of TechnologyJournal Article20170209In 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.https://toc.ui.ac.ir/article_21474_4d4d47f1c8519b0e97e542738798fb06.pdfUniversity of IsfahanTransactions on Combinatorics2251-86577120180301PD-sets for codes related to flag-transitive symmetric designs37502161510.22108/toc.2017.21615ENDeanCrnkovicDepartment of Mathematics, University of Rijeka, Radmile Matječić 2, 51000 Rijeka, CroatiaNinaMostaracDepartment of Mathematics, University of Rijeka, Rijeka, CroatiaJournal Article20170604For any prime $p$ let $C_p(G)$ be the $p$-ary code spanned by the rows of the incidence matrix $G$ of a graph $\Gamma$. Let $\Gamma$ be the incidence graph of a flag-transitive symmetric design $D$. We show that any flag-transitive automorphism group of $D$ can be used as a PD-set for full error correction for the linear code $C_p(G)$ (with any information set). It follows that such codes derived from flag-transitive symmetric designs can be decoded using permutation decoding. In that way to each flag-transitive symmetric $(v, k, \lambda)$ design we associate a linear code of length $vk$ that is permutation decodable. PD-sets obtained in the described way are usually of large cardinality. By studying codes arising from some flag-transitive symmetric designs we show that smaller PD-sets can be found for specific information sets.https://toc.ui.ac.ir/article_21615_538caf5ff8ba2437eee5ab750d6dce2a.pdf