### فهرست مطالب • Volume:3 Issue:2, 2014
• تاریخ انتشار: 1393/02/15
• تعداد عناوین: 8
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Let \$A\$ be a non-trivial abelian group and \$A^{*}=Asetminus {0}\$. A graph \$G\$ is said to be \$A\$-magic graph if there exists a labeling \$l:E(G)rightarrow A^{*}\$ such that the induced vertex labeling \$l^{+}:V(G)rightarrow A\$, define by \$\$l^+(v)=sum_{uvin E(G)} l(uv)\$\$ is a constant map.The set of all constant integers such that \$sum_{uin N(v)} l(uv)=c\$, for each \$vin N(v)\$, where \$N(v)\$ denotes the set of adjacent vertices to vertex \$v\$ in \$G\$, is called the index set of \$G\$ and denoted by \${rm In}_{A}(G).\$ In this paper we determine the index set of certain planar graphs for \$mathbb{Z}_{h}\$, where \$hin mathbb{N}\$, such as wheels and fans.
Keywords: Index Set, Magic, Zero, Sum, Null Set
• Jaisankar Senbagamalar, Jayapal Baskar Babujee, Ivan Gutman Pages 11-15
Let \$G\$ be an \$(n,m)\$-graph. We say that \$G\$ has property \$(ast)\$ if for every pair of its adjacent vertices \$x\$ and \$y\$, there exists a vertex \$z\$, such that \$z\$ is not adjacent to either \$x\$ or \$y\$. If the graph \$G\$ has property \$(ast)\$, then its complement \$overline G\$ is connected, has diameter 2, and its Wiener index is equal to \$binom{n}{2}+m\$, i.e., the Wiener index is insensitive of any other structural details of the graph \$G\$. We characterize numerous classes of graphs possessing property \$(ast)\$, among which are trees, regular, and unicyclic graphs.
Keywords: distance (in graphs), Wiener index, complement (of graph)
• Alireza Abdollahi, Shahrooz Janbaz Pages 17-20
Let \$n\$ be any positive integer, the friendship graph \$F_n\$ consists of \$n\$ edge-disjoint triangles that all of them meeting in one vertex. A graph \$G\$ is called cospectral with a graph \$H\$ if their adjacency matrices have the same eigenvalues. Recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if \$G\$ is any graph cospectral with \$F_n\$ (\$nneq 16\$), then \$Gcong F_n\$. In this note, we give a proof of a special case of the latter: Any connected graph cospectral with \$F_n\$ is isomorphic to \$F_n\$.Our proof is independent of ones given in href{http://arxiv.org/pdf/1310.6529v1.pdf http://arxiv.org/pdf/1310.6529v1.pdf} and the proofs are based on our recent results iven in [{em Trans. Comb.}, {bf 2} no. 4 (2013) 37-52.] using an upper bound for the argest eigenvalue of a connected graph given in {em J. Combinatorial Theory Ser. B} {bf 81} (2001) 177-183.].
• Zhaoyang Luo, Jianliang Wu Pages 21-29
Let \$G\$ be a connected graph. The multiplicative Zagreb eccentricity indices of \$G\$ are defined respectively as \${bf Pi}_1^*(G)=prod_{vin V(G)}varepsilon_G^2(v)\$ and \${bf Pi}_2^*(G)=prod_{uvin E(G)}varepsilon_G(u)varepsilon_G(v)\$, where \$varepsilon_G(v)\$ is the eccentricity of vertex \$v\$ in graph \$G\$ and \$varepsilon_G^2(v)=(varepsilon_G(v))^2\$. In this paper, we present some bounds of the multiplicative Zagreb eccentricity indices of Cartesian product graphs by means of some invariants of the factors and supply some exact expressions of \${bf Pi}_1^*\$ and \${bf Pi}_2^*\$ of some composite graphs, such as the join, disjunction, symmetric difference and composition of graphs, respectively.
Keywords: Multiplicative Zagreb eccentricity indices, composite operations, Cartesian product
• Gholamreza Omidi, Khosro Tajbakhsh Pages 31-33
For a given hypergraph \$H\$ with chromatic number \$chi(H)\$ and with no edge containing only one vertex, it is shown that the minimum number \$l\$ for which there exists a partition (also a covering) \${E_1,E_2,ldots,E_l}\$ for \$E(H)\$, such that the hypergraph induced by \$E_i\$ for each \$1leq ileq l\$ is \$k\$-colorable, is \$lceil log_{k} chi(H) rceil\$.
• Maryam Atapour, Sepideh Norouzian, Seyed Mahmoud Sheikholeslami Pages 35-44
A function \$f:V(G)rightarrow {-1,0,1}\$ is a {em minus dominating function} if for every vertex \$vin V(G)\$, \$sum_{uin N[v]}f(u)ge 1\$. A minus dominating function \$f\$ of \$G\$ is called a {em global minus dominating function} if \$f\$ is also a minus dominating function of the complement \$overline{G}\$ of \$G\$. The {em global minus domination number} \$gamma_{g}^-(G)\$ of \$G\$ is defined as \$gamma_{g}^-(G)=min{sum_{vin V(G)} f(v)mid f mbox{is a global minus dominating function of} G}\$. In this paper we initiate the study of global minus domination number in graphs and we establish lower and upper bounds for the global minus domination number.
Keywords: minus dominating function, minus domination number, global minus dominating function, global minus domination number
• R. Lakshmi, S. Vidhyapriya Pages 45-49
A kernel J of a digraph D is an independent set of vertices of D such that for every vertex w ∈ V (D) ∖ J there exists an arc from w to a vertex in J. In this paper, among other results, a characterization of 2 -regular circulant digraph having a kernel is obtained. This characterization is a partial solution to the following problem: Characterize circulant digraphs which have kernels; it appeared in the book Digraphs - theory, algorithms and applications, Second Edition, Springer-Verlag, 2009, by J. Bang-Jensen and G. Gutin.
Keywords: Kernel, Symmetric Digraphs, Circulant Digraph
• Anwar Alwardi, Karam Ebadi, Martin Manrique, Nsndappa Soner Pages 51-63
Given a graph \$G = (V,E)\$, a dominating set \$D subseteq V\$ is called a semi-strongsplit dominating set of \$G\$ if \$|V setminus D| geq 1\$ and the maximum degree of the subgraph induced by \$V setminus D\$ is 1. The minimum cardinality of a semi-strong split dominating set (SSSDS) of G is the semi-strong split domination number of G, denoted \$gamma_{sss}(G)\$. In this work, we introduce the concept and prove several results regarding it.
Keywords: split domination, strong split domination, tree