University of IsfahanTransactions on Combinatorics2251-86571120120301Note on edge distance-balanced graphs168610.22108/toc.2012.86ENM.TavakoliH.Yousefi-AzariAli RezaAshrafiJournal Article20110803Edge distance-balanced graphs are graphs in which for every edge $e = uv$ the number of edges closer to vertex $u$ than to vertex $v$ is equal to the number of edges closer to $v$ than to $u$. In this paper, we study this property under some graph operations.https://toc.ui.ac.ir/article_86_41c9776f55f12410740dfdfe4fa080ec.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571120120301$k$-Tuple total domination and mycieleskian graphs71333310.22108/toc.2012.333ENAdelP. KazemiUMA
(University of Mohaghegh Ardabili)Journal Article20111101Let $k$ be a positive integer. A subset $S$ of $V(G)$ in a graph $G$ is a $k$-tuple total dominating set of $G$ if every vertex of $G$ has at least $k$ neighbors in $S$. The $k$-tuple total domination number $\gamma _{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$. In this paper for a given graph $G$ with minimum degree at least $k$, we find some sharp lower and upper bounds on the $k$-tuple total domination number of the $m$ -Mycieleskian graph $\mu _{m}(G)$ of $G$ in terms on $k$ and $\gamma _{\times k,t}(G)$. Specially we give the sharp bounds $\gamma _{\times k,t}(G)+1$ and $\gamma _{\times k,t}(G)+k$ for $\gamma _{\times k,t}(\mu _1(G))$, and characterize graphs with $\gamma _{\times k,t}(\mu _1(G))=\gamma _{\times k,t}(G)+1$.https://toc.ui.ac.ir/article_333_1c8f4a659abb4fec82b4bd7691dba941.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571120120301Product-cordial index and friendly index of regular graphs152048210.22108/toc.2012.482ENWai CheeShiuHong Kong Baptist UniversityKwongHarrisState University of New York at FredoniaJournal Article20111123Let $G=(V,E)$ be a connected simple graph. A labeling $f: V\to Z_2$ induces two edge labelings $f^+, f^*: E \to Z_2$ defined by $f^+(xy) = f(x)+f(y)$ and $f^*(xy) = f(x)f(y)$ for each $xy \in E$. For $i \in Z_2$, let $v_f(i) = |f^{-1}(i)|$, $e_{f^+}(i) = |(f^{+})^{-1}(i)|$ and $e_{f^*}(i) = |(f^*)^{-1}(i)|$. A labeling $f$ is called friendly if $|v_f(1)-v_f(0)| \le 1$. For a friendly labeling $f$ of a graph $G$, the friendly index of $G$ under $f$ is defined by $i^+_f(G) = e_{f^+}(1)-e_{f^+}(0)$. The set $\{i^+_f(G)\;|\;f \mbox{ is a friendly labeling of} G\}$ is called the full friendly index set of $G$. Also, the product-cordial index of $G$ under $f$ is defined by $i^*_f(G) = e_{f^*}(1)-e_{f^*}(0)$. The set $\{i^*_f(G)\;|\;f \mbox{ is a friendly labeling of} G\}$ is called the full product-cordial index set of $G$. In this paper, we find a relation between the friendly index and the product-cordial index of a regular graph. As applications, we will determine the full product-cordial index sets of torus graphs which was asked by Kwong, Lee and Ng in 2010; and those of cycles.https://toc.ui.ac.ir/article_482_0f0636556a2c2f345a1e9f8e54e0934d.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571120120301Minimal, vertex minimal and commonality minimal CN-dominating graphs212949710.22108/toc.2012.497ENAnwar SalehAlwardiUniversity of MysoreN. D.SonerUniversity of MysoreJournal Article20111119We define minimal CN-dominating graph $\mathbf {MCN}(G)$, commonality minimal CN-dominating graph $\mathbf {CMCN}(G)$ and vertex minimal CN-dominating graph $\mathbf {M_{v}CN}(G)$, characterizations are given for graph $G$ for which the newly defined graphs are connected. Further serval new results are developed relating to these graphs.https://toc.ui.ac.ir/article_497_907cf6e71495c5344706494bd6c58cf4.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571120120301A note on star coloring of central graph of bipartite graph and corona graph of complete graph with path and cycle313463610.22108/toc.2012.636ENV. J.VernoldAnna University of Technology TirunelveliM.VenkatachalamR.V.S Faculty of EngineeringJournal Article20120219In this paper, we find the star chromatic number of central graph of complete bipartite graph and corona graph of complete graph with path and cycle.https://toc.ui.ac.ir/article_636_96d78bc268249e8f330d8a4596ed626b.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571120120301Complexity indices for the travelling salesman problem and data mining354372310.22108/toc.2012.723ENDragosCvetkovićJournal Article20111122 we extend our previous work on complexity indices for the travelling salesman problem (TSP), summarized in cite{CvCK3}, using graph spectral techniques of data mining. A complexity index is an invariant of an instance $I$ by which we can predict the execution time of an exact algorithm for TSP for $I$. We consider the symmetric travelling salesman problem with instances $I$ represented by complete graphs $G$ with distances between vertices (cities) as edge weights (lengths). Intuitively, the hardness of an instance $G$ depends on the distribution of short edges within $G$. Therefore we consider some short edge subgraphs of $G$ (minimal spanning tree, critical connected subgraph, and several others) as non-weighted graphs and several their invariants as potential complexity indices. Here spectral invariants (e.g. spectral radius of the adjacency matrix) play an important role since, in general, there are intimate relations between eigenvalues and the structure of a graph. Since hidden details of short edge subgraphs really determine the hardness of the instance, we use techniques of data mining to find them. In particular, spectral clustering algorithms are used including information obtained from the spectral gap in Laplacian spectra of short edge subgraphs.https://toc.ui.ac.ir/article_723_a83c593b06f18b6cdb9a9a465d56305d.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571120120301On the total domatic number of regular graphs455176010.22108/toc.2012.760ENH.AramAzarbaijan University of Tarbiat MoallemS. M.SheikholeslamiAzarbaijan University of Tarbiat MoallemL.VolkmannRWTH-Aachen UniversityJournal Article20120112A set $S$ of vertices of a graph $G=(V,E)$ without isolated vertex is a total dominating set if every vertex of $V(G)$ is adjacent to some vertex in $S$. The <em>total domatic number</em> of a graph $G$ is the maximum number of total dominating sets into which the vertex set of $G$ can be partitioned. We show that the total domatic number of a random $r$-regular graph is almost surely at most $r-1$, and that for 3-regular random graphs, the total domatic number is almost surely equal to 2. We also give a lower bound on the total domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the total domatic number of an $r$-regular graph is at least $r/(3\ln(r))$.https://toc.ui.ac.ir/article_760_362cc9c41ad1def424bd149103450c49.pdf