University of IsfahanTransactions on Combinatorics2251-86572220130601On the complexity of the colorful directed paths in vertex coloring of digraphs17284010.22108/toc.2013.2840ENS.SaqaeeyanAbadan Branch, Islamic Azad UniversityEsmaeilMollaahmadiSharif University of Technology .AliDehghanAmirkabir University of Technology, Tehran, IranJournal Article20121016The colorful paths and rainbow paths have been considered by several authors. A colorful directed path in a digraph $G$ is a directed path with $\chi(G)$ vertices whose colors are different. A $v$-colorful directed path is such a directed path, starting from $v$. We prove that for a given $3$-regular triangle-free digraph $G$ determining whether there is a proper $\chi(G)$-coloring of $G$ such that for every $v \in V (G)$, there exists a $v$-colorful directed path is $ \mathbf{NP} $-complete.https://toc.ui.ac.ir/article_2840_6a5c24f33fd5a66915e473e2c44ca4aa.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572220130601Convolutional cylinder-type block-circulant cycle codes917284810.22108/toc.2013.2848ENMohammadGholamiShahrekord UniversityMehdiSamadiehIsfahan Mathematics HouseJournal Article20121029In this paper, we consider a class of column-weight two quasi-cyclic low-density parity check codes in which the girth can be large enough, as an arbitrary multiple of 8. Then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. Finally, we show that the free distance of the convolutional codes is equal to the minimum distance of
their block counterparts.https://toc.ui.ac.ir/article_2848_9c85537997b5b347da497a3d38139266.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572220130601On schemes originated from Ferrero pairs1926286910.22108/toc.2013.2869ENHosseinMoshtaghDepartment of
Mathematics, K. N. Toosi University of Technology, AmirRahnamai BarghiK. N. Toosi university of Technology University, Tehran-Iran.Journal Article20130519The Frobenius complement of a given Frobenius group acts on its kernel. The scheme which is arisen from the orbitals of this action is called Ferrero pair scheme. In this paper, we show that the fibers of a Ferrero pair scheme consist of exactly one singleton fiber and every two fibers with more than one point have the same cardinality. Moreover, it is shown that the restriction of a Ferrero pair scheme on each fiber is isomorphic to a regular scheme. Finally, we prove that for any prime $p$, there exists a Ferrero pair $p$-scheme, and if $p> 2$, then the Ferrero pair $p$-schemes of the same rank are all isomorphic.https://toc.ui.ac.ir/article_2869_9eaaa1dafa631de15cb2c9f513a98e5c.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572220130601On the number of cliques and cycles in graphs2733287210.22108/toc.2013.2872ENMasoudAriannejadUniversity of zanjanMojganEmamiDepartment of Mathematics, University of ZanjanJournal Article20121203We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let $G$ be a graph and let $\overline {G}$ be its complement, then given the chromatic polynomial of $\overline {G}$, we give a recursive method to compute the number of cliques of $G$. Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$. In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$.https://toc.ui.ac.ir/article_2872_183b76bba4970596525b994ca1ef4997.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572220130601Probabilistic analysis of the first Zagreb index3540288110.22108/toc.2013.2881ENRaminKazemiDepartment of statistics, Imam Khomeini International University, QazvinJournal Article20130420In this paper we study the first Zagreb index in bucket recursive trees containing buckets with variable capacities. This model was introduced by Kazemi in 2012. We obtain the mean and variance of the first Zagreb index and <br />introduce a martingale based on this quantity.https://toc.ui.ac.ir/article_2881_e4d82056fd1c36fd883f73551fe4a60f.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572220130601On the spectra of reduced distance matrix of dendrimers4146289010.22108/toc.2013.2890ENAbbasHeydaristaffJournal Article20130223Let $G$ be a simple connected graph and $\{v_1,v_2,\ldots, v_k\}$ be the set of pendent (vertices of degree one) vertices of $G$. The reduced distance matrix of $G$ is a square matrix whose $(i,j)$-entry is the topological distance between $v_i$ and $v_j$ of $G$. In this paper, we obtain the spectrum of the reduced distance matrix of regular dendrimers.https://toc.ui.ac.ir/article_2890_73b2bef50330eda9f10bd13a01debf78.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572220130601Modular chromatic number of $C_m \square P_n$4772294310.22108/toc.2013.2943ENN.ParamaguruAnnamalai UniversityR.SampathkumarAnnamalai UniversityJournal Article20130125A modular $k\!$-coloring, $k\ge 2,$ of a graph $G$ is a coloring of the vertices of $G$ with the elements in $\mathbb{Z}_k$ having the property that for every two adjacent vertices of $G,$ the sums of the colors of their neighbors are different in $\mathbb{Z}_k.$ The minimum $k$ for which $G$ has a modular $k\!$-coloring is the modular chromatic number of $G.$ Except for some special cases, modular chromatic number of $C_m\square P_n$ is determined.https://toc.ui.ac.ir/article_2943_5fa15d9e433e3cf4e5d1849c0214c4df.pdf