University of IsfahanTransactions on Combinatorics2251-86572420131201Roman game domination subdivision number of a graph112334110.22108/toc.2013.3341ENJafarAmjadiAzarbaijan Shahid Madani UniversityHosseinKaramiAzarbaijan Shahid Madani UniversitySeyed MahmoudSheikholeslamiAzarbaijan University of Tarbiat MoallemLutzVolkmannRWTH-Aachen UniversityJournal Article20130620A <em>Roman dominating function</em> on a graph $G = (V,E)$ is a function $f : V\longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. The <em>weight</em> of a Roman dominating function is the value $w(f)=\sum_{v\in V}f(v)$. The Roman domination number of a graph $G$, denoted by $\gamma_R(G)$, equals the minimum weight of a Roman dominating function on G. The Roman game domination subdivision number of a graph $G$ is defined by the following game. Two players $\mathcal D$ and $\mathcal A$, $\mathcal D$ playing first, alternately mark or subdivide an edge of $G$ which is not yet marked nor subdivided. The game ends when all the edges of $G$ are marked or subdivided and results in a new graph $G'$. The purpose of $\mathcal D$ is to minimize the Roman domination number $\gamma_R(G')$ of $G'$ while $\mathcal A$ tries to maximize it. If both $\mathcal A$ and $\mathcal D$ play according to their optimal strategies, $\gamma_R(G')$ is well defined. We call this number the {\em Roman game domination subdivision number} of $G$ and denote it by $\gamma_{Rgs}(G)$. In this paper we initiate the study of the Roman game domination subdivision number of a graph and present sharp bounds on the Roman game domination subdivision number of a tree.https://toc.ui.ac.ir/article_3341_b03cc8118595dcee034dcf7f43bede8d.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572420131201Reciprocal degree distance of some graph operations1324350610.22108/toc.2013.3506ENKannanPattabiramanAnnamalai UniversityM.VijayaragavanThiruvalluvar College of Engineering and TechnologyJournal Article20130820The reciprocal degree distance (RDD), defined for a connected graph $G$ as vertex-degree-weighted sum of the reciprocal distances, that is, $RDD(G) =\sum\limits_{u,v\in V(G)}\frac{d_G(u) + d_G(v)}{d_G(u,v)}.$ The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. In this paper, we present exact formulae for the reciprocal degree distance of join, tensor product, strong product and wreath product of graphs in terms of other graph invariants including the degree distance, Harary index, the first Zagreb index and first Zagreb coindex. Finally, we apply some of our results to compute the reciprocal degree distance of fan graph, wheel graph, open fence and closed fence graphs.https://toc.ui.ac.ir/article_3506_bfc02f2003611e70f89d3bbb16913cd1.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572420131201Graph theoretical methods to study controllability and leader selection for dead-time systems2536361610.22108/toc.2013.3616ENMajdeddinNajafiAvionics Research InstituteFaridShaikholeslamElectrical & Computer Engineering DepartmentJournal Article20130913In this article a graph theoretical approach is employed to study some specifications of dynamic systems with time delay in the inputs and states, such as structural controllability and observability. First, the zero and non-zero parameters of a proposed system have been determined, next the general structure of the system is presented by a graph which is constructed by non-zero parameters. The structural controllability and observability of the system is investigated using the corresponding graph. Our results are expressed for multi-agents systems with dead-time. As an application we find a minimum set of leaders to control a given multi-agent system.https://toc.ui.ac.ir/article_3616_b2a99bce0bfaf49e3d546e59ee0937c0.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572420131201Graphs cospectral with a friendship graph or its complement3752362110.22108/toc.2013.3621ENAlirezaAbdollahiUniversity of IsfahanShahroozJanbazUniversity of IsfahanMohammad RezaOboudiUniversity of IsfahanJournal Article20130720Let $n$ be any positive integer and $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length $4$ or $5$, then $G\cong F_n$. Moreover if $G$ is connected and planar then $G\cong F_n$. All but one of connected components of $G$ are isomorphic to $K_2$. The complement $\overline{F_n}$ of the friendship graph is determined by its adjacency eigenvalues, that is, if $\overline{F_n}$ is cospectral with a graph $H$, then $H\cong \overline{F_n}$.https://toc.ui.ac.ir/article_3621_0bcd0f5df9a893e748683a0325f8cac6.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572420131201Directionally $n$-signed graphs-III: the notion of symmetric balance5362365810.22108/toc.2013.3658ENP.Siva KotaReddyDept. of Mathematics, Siddaganga Institute of Technology, B.H.Road,Tumkur-572103, India.U. K.MisraBerhampur UniversityJournal Article20130818Let $G=(V, E)$ be a graph. By \emph{directional labeling (or d-labeling)} of an edge $x=uv$ of $G$ by an ordered $n$-tuple $(a_1,a_2,\dots,a_n)$, we mean a labeling of the edge $x$ such that we consider the label on $uv$ as $(a_1,a_2,\dots,a_n)$ in the direction from $u$ to $v$, and the label on $x$ as $(a_{n},a_{n-1},\dots,a_1)$ in the direction from $v$ to $u$. In this paper, we study graphs, called \emph{(n,d)-sigraphs}, in which every edge is $d$-labeled by an $n$-tuple $(a_1,a_2,\dots,a_n)$, where $a_k \in \{+,-\}$, for $1\leq k \leq n$. In this paper, we give different notion of balance: symmetric balance in a $(n,d)$-sigraph and obtain some characterizations.https://toc.ui.ac.ir/article_3658_afa2e9837ed9f7e3782ebee733da0db9.pdfUniversity of IsfahanTransactions on Combinatorics2251-86572420131201Full friendly index sets of slender and flat cylinder graphs6380367810.22108/toc.2013.3678ENWai CheeShiuHong Kong Baptist UniversityMan-HoHoHong Kong Baptist UniversityJournal Article20130904Let $G=(V,E)$ be a connected simple graph. A labeling $f:V \to Z_2$ induces an edge labeling $f^*:E \to Z_2$ defined by $f^*(xy)=f(x)+f(y)$ for each $xy \in E$. For $i \in Z_2$, let $v_f(i)=|f^{-1}(i)|$ and $e_f(i)=|f^{*-1}(i)|$. A labeling $f$ is called friendly if $|v_f(1)-v_f(0)|\le 1$. The full friendly index set of $G$ consists all possible differences between the number of edges labeled by 1 and the number of edges labeled by 0. In recent years, full friendly index sets for certain graphs were studied, such as tori, grids $P_2\times P_n$, and cylinders $C_m\times P_n$ for some $n$ and $m$. In this paper we study the full friendly index sets of cylinder graphs $C_m\times P_2$ for $m\geq 3$, $C_m\times P_3$ for $m\geq 4$ <br />and $C_3\times P_n$ for $n\geq 4$. The results in this paper complement the existing results <br />in literature, so the full friendly index set of cylinder graphs are completely determined.https://toc.ui.ac.ir/article_3678_3225bdc21d140f967414ef14f7247734.pdf