University of IsfahanTransactions on Combinatorics2251-86575420161201Cacti with extremal PI Index181478610.22108/toc.2016.14786ENChunxiangWangCentral China Normal UniversityShaohuiWangUniversity of MississippiBingWeiUniversity of MississippiJournal Article20160224The vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the greatest and smallest vertex PI indices mong all cacti with a fixed number of vertices. As a consequence, we obtain the sharp bounds with corresponding extremal cacti and extend a known result.https://toc.ui.ac.ir/article_14786_f95e820e8bf0d1325600f95c8a3d7a24.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575420161201Some results on the comaximal ideal graph of a commutative ring9201504710.22108/toc.2016.15047ENHamid RezaDorbidiUniversity of Jiroft,Jiroft, Kerman, IranRaoufehManaviyatPayame Noor University, Tehran, IranJournal Article20150727Let $R$ be a commutative ring with unity. The comaximal ideal graph of $R$, denoted by $\mathcal{C}(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 +I_2 = R$. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of $\mathcal{C}(R)$ for a ring $R$ is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some $\mathcal{C}(\mathbb{Z}_n)$.https://toc.ui.ac.ir/article_15047_e2760f540dc55e62152260c257848270.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575420161201On the new extension of distance-balanced graphs21341504810.22108/toc.2016.15048ENMortezaFaghaniDepartment of Mathematics, Payame Noor University, P.O.Box 9395-3697, Tehran, IranEhsanPourhadiSchool of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, IranHassanKharaziDepartment of Mathematics and Statistics, Imam Hossein University, Tehran, IranJournal Article20160203In this paper, we initially introduce the concept of $n$-distance-balanced property which is considered as the generalized concept of distance-balanced property. In our consideration, we also define the new concept locally regularity in order to find a connection between $n$-distance-balanced graphs and their lexicographic product. Furthermore, we include a characteristic method which is practicable and can be used to classify all graphs with $i$-distance-balanced properties for $ i=2,3 $ which is also relevant to the concept of total distance. Moreover, we conclude a connection between distance-balanced and 2-distance-balanced graphs.https://toc.ui.ac.ir/article_15048_3968109258ac5aaddf5a16c03fc677d5.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575420161201Extremal tetracyclic graphs with respect to the first and second Zagreb indices35551287810.22108/toc.2016.12878ENNaderHabibiuniversity of Ayatollah Al-ozmaTayebehDehghan ZadehUniversity of KashanAli RezaAshrafiUniversity of KashanJournal Article20160113The first Zagreb index, $M_1(G)$, and second Zagreb index, $M_2(G)$, of the graph $G$ is defined as $M_{1}(G)=\sum_{v\in V(G)}d^{2}(v)$ and $M_{2}(G)=\sum_{e=uv\in E(G)}d(u)d(v),$ where $d(u)$ denotes the degree of vertex $u$. In this paper, the first and second maximum values of the first and second Zagreb indices in the class of all $n-$vertex tetracyclic graphs are presented.https://toc.ui.ac.ir/article_12878_b7525583ad7d958b2f5cb6c2d9eabfdb.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575420161201Congruences from $q$-Catalan Identities57672035810.22108/toc.2016.20358ENQingZouDepartment of Mathematics, The University of IowaJournal Article20160213In this paper, by studying three $q$-Catalan identities given by Andrews, we arrive at a certain number of congruences. These congruences are all modulo $\Phi_n(q)$, the $n$-th cyclotomic polynomial or the related functions and modulo $q$-integers.https://toc.ui.ac.ir/article_20358_742244d2cadb0585b9b1cc7a3cde94c5.pdf