University of IsfahanTransactions on Combinatorics2251-86578420191201Coloring problem of signed interval graphs192384910.22108/toc.2019.108880.1541ENFarzanehRamezaniFaculty of Mathematics, K. N. Toosi University of Technology, Tehran, IranJournal Article20180303A signed graph $(G,sigma)$ is a graph together with an assignment of signs ${+,-}$ to its edges where $sigma$ is the subset of its negative edges. There are a few variants of coloring and clique problems of signed graphs, which have been studied. An initial version known as vertex coloring of signed graphs is defined by Zaslavsky in $1982$. Recently Naserasr et. al., in [R. Naserasr, E. Rollova and E. Sopena, Homomorphisms of signed graphs,<em> J. Graph Theory</em>, <strong>79</strong> (2015) 178--212, have defined signed chromatic and signed clique numbers of signed graphs. In this paper we consider the latter mentioned problems for signed interval graphs. We prove that the coloring problem of signed interval graphs is NP-complete whereas their ordinary coloring problem (the coloring problem of interval graphs) is in P. Moreover we prove that the signed clique problem of a signed interval graph can be solved in polynomial time. We also consider the complexity of further related problems.<br /> <br /><br />http://toc.ui.ac.ir/article_23849_1554f4e5c9d7ae542ab410c68865a403.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578420191201Elliptic root systems of type $A_1$, a combinatorial study11212402310.22108/toc.2019.117338.1648ENZahraKharaghaniDepartment of mathematics, University of Isfahan, Isfahan, IranJournal Article20190529We consider some combinatorics of elliptic root systems of type $A_1$. In particular, with respect to a fixed reflectable base, we give a precise description of the positive roots in terms of a ``positivity'' theorem. Also the set of reduced words of the corresponding Weyl group is precisely described. These then lead to a new characterization of the core of the corresponding Lie algebra, namely we show that the core is generated by positive root spaces.http://toc.ui.ac.ir/article_24023_773c7ef2a4751d9ad0477a8e0cf7b337.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578420191201Some subgroups of $mathbb{F}_q^*$ and explicit factors of $x^{2^nd}-1inmathbb{F}_q[x]$23332413910.22108/toc.2019.114742.1612ENManjitSinghDepartment of
Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal-131039, Sonepat, India0000-0003-3351-7287Journal Article20181225Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $mathcal{O}_q$ be the set of all odd order elements of $mathbb{F}_q^*$. Then $mathcal{O}_q$ turns up as a subgroup of $mathcal{S}_q$. In this paper, we show that $mathcal{O}_q=langle4rangle$ if $q=2t+1$ and, $mathcal{O}_q=langle trangle $ if $q=4t+1$, where $q$ and $t$ are odd primes. Further, we determine the coefficients of irreducible factors of $x^{2^nt}-1$ using generators of these special subgroups of $mathbb{F}_q^*$http://toc.ui.ac.ir/article_24139_c7f69a9cdc05b87cd6175f9ab6f48e5c.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578420191201Generalized Zagreb index of product graphs35482402410.22108/toc.2019.116001.1625ENMahdiehAzariKazerun Branch, Islamic Azad UniversityJournal Article20190311The generalized Zagreb index is an extension of both ordinary and variable Zagreb indices. In this paper, we present exact formulae for the values of the generalized Zagreb index for product graphs. Results are applied to some graphs of general and chemical interest such as nanotubes and nanotori.http://toc.ui.ac.ir/article_24024_bcdf8c66d8da47e8cdf5362669ab0f75.pdfUniversity of IsfahanTransactions on Combinatorics2251-86578420191201Some upper bounds for the signless Laplacian spectral radius of digraphs49602424510.22108/toc.2019.105894.1515ENWeigeXiDepartment of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, P.R.ChinaLigongWangNorthwestern Polytechnical UniversityJournal Article20170812Let $G=(V(G),E(G))$ be a digraph without loops and multiarcs, where $V(G)={v_1,v_2,$ $ldots,v_n}$ and $E(G)$ are the vertex set and the arc set of $G$, respectively. Let $d_i^{+}$ be the outdegree of the vertex $v_i$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)=textrm{diag}(d_1^{+},d_2^{+},ldots,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. The spectral radius of $Q(G)$ is called the signless Laplacian spectral radius of $G$, denoted by $q(G)$. In this paper, some upper bounds for $q(G)$ are obtained. Furthermore, some upper bounds on $q(G)$ involving outdegrees and the average 2-outdegrees of the vertices of $G$ are also derived.http://toc.ui.ac.ir/article_24245_2f33bf10a12951107f529f5d0ee19c1b.pdf