University of IsfahanTransactions on Combinatorics2251-86579120200301A Linear Algorithm for Computing $gamma_{_{[1,2]}}$-set in Generalized Series-Parallel Graphs1242418510.22108/toc.2019.105482.1509ENPouyehSharifaniDepartment of Computer Science, Yazd University, Yazd, IranMohammad RezaHooshmandaslDepartment of Computer Science, Yazd University, Yazd, Iran0000-0002-3834-3610Journal Article20170720For a graph $G=(V,E)$, a set $S subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v in V setminus S$ is dominated by at most two vertices of $S$, i.e. $1 leq vert N(v) cap S vert leq 2$. Moreover a set $S subseteq V$ is a total $[1,2]$-set if for each vertex of $V$, it is the case that $1 leq vert N(v) cap S vert leq 2$. The $[1,2]$-domination number of $G$, denoted $gamma_{[1,2]}(G)$, is the minimum number of vertices in a $[1,2]$-set. Every $[1,2]$-set with cardinality of $gamma_{[1,2]}(G)$ is called a $gamma_{[1,2]}$-set. Total $[1,2]$-domination number and $gamma_{t[1,2]}$-sets of $G$ are defined in a similar way. This paper presents a linear time algorithm to find a $gamma_{[1,2]}$-set and a $gamma_{t[1,2]}$-set in generalized series-parallel graphs.https://toc.ui.ac.ir/article_24185_5d3db88464af44f66f4256faa00162d7.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579120200301On a generalization of Leray simplicial complexes25302425710.22108/toc.2019.119856.1682ENSiamakYassemiUniversity of TehranJournal Article20191030We define a refinement of the notion of Leray simplicial complexes and study its properties. Moreover, we translate some of our results to the language of commutative algebra.https://toc.ui.ac.ir/article_24257_f994299644df0e2b4c4dc0c3520742e0.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579120200301Bounds for metric dimension and defensive $k$-alliance of graphs under deleted lexicographic product31392408110.22108/toc.2019.115674.1622ENKinkar ChandraDasSungkyunkwan UniversityMostafaTavakoliDepartment of Applied Mathematics, Faculty of Mathematical Sciences,
Ferdowsi University of MashhadJournal Article20190223Metric dimension and defensive $k$-alliance number are two distance-based graph invariants which have applications in robot navigation, quantitative analysis of secondary RNA structures, national defense and fault-tolerant computing. In this paper, some bounds for metric dimension and defensive $k$-alliance of deleted lexicographic product of graphs are presented. We also show that the bounds are sharp.https://toc.ui.ac.ir/article_24081_5407a5ecbac3bfcf53a896f76922b370.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579120200301Nilpotent graphs of skew polynomial rings over non-commutative rings41482432110.22108/toc.2019.117529.1651ENMohammad JavadNikmehrK.N.Toosi UniversityAbdolrezaAzadiK. N. Toosi University of TechnologyJournal Article20190608Let $R$ be a ring and $alpha$ be a ring endomorphism of $R$. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set $Z_N(R)^*$, and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent, where $Z_N(R)={xin R;|; xy; rm{is; nilpotent,;for; some}; yin R^*}.$ In this article, we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;alpha]$ and the graph-theoretical properties of its nilpotent graph $Gamma_N(R[x;alpha])$. It is shown that if $R$ is a symmetric and $alpha$-compatible with exactly two minimal primes, then $diam(Gamma_N(R[x,alpha]))=2$. Also we prove that $Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $Z_2timesZ_2$.https://toc.ui.ac.ir/article_24321_df4b13199d579e2cea343d25bcb434d6.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579120200301Transitive distance-regular graphs from linear groups $L(3,q)$, $q = 2,3,4,5$49602440110.22108/toc.2020.116255.1630ENAndreaSvobDepartment of Mathematics, University of Rijeka, CroatiaJournal Article20190401In this paper we classify distance-regular graphs, including strongly regular graphs, admitting a transitive action of the linear groups $L(3,2)$, $L(3,3)$, $L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15. We give details about constructed graphs. In addition, we construct self-orthogonal codes from distance-regular graphs obtained in this paper.https://toc.ui.ac.ir/article_24401_2717675239d9099d8570ae8b307013ae.pdf