University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2
4
2013
12
01
Roman game domination subdivision number of a graph
1
12
EN
Jafar
Amjadi
Azarbaijan Shahid Madani University
j-amjadi@azaruniv.edu
Hossein
Karami
Azarbaijan Shahid Madani University
h.karami_math@yahoo.com
Seyed Mahmoud
Sheikholeslami
Azarbaijan University of Tarbiat Moallem
s.m.sheikholeslami@azaruniv.edu
Lutz
Volkmann
RWTH-Aachen University
volkm@math2.rwth-aachen.de
10.22108/toc.2013.3341
A Roman dominating function on a graph $G = (V,E)$ is a function $f : Vlongrightarrow {0, 1, 2}$ 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 weight of a Roman dominating function is the value $w(f)=sum_{vin V}f(v)$. 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.
Roman domination number,Roman game domination subdivision number,tree
http://toc.ui.ac.ir/article_3341.html
http://toc.ui.ac.ir/article_3341_b03cc8118595dcee034dcf7f43bede8d.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2
4
2013
12
01
Reciprocal degree distance of some graph operations
13
24
EN
Kannan
Pattabiraman
Annamalai University
pramank@gmail.com
M.
Vijayaragavan
Thiruvalluvar College of Engineering and Technology
mvragavan09@gmail.com
10.22108/toc.2013.3506
The reciprocal degree distance (RDD), defined for a connected graph $G$ as vertex-degree-weighted sum of the reciprocal distances, that is, $RDD(G) =sumlimits_{u,vin V(G)}frac{d_G(u) + d_G(v)}{d_G(u,v)}.$ 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.
Reciprocal degree distance,Harary index,Graph operations
http://toc.ui.ac.ir/article_3506.html
http://toc.ui.ac.ir/article_3506_bfc02f2003611e70f89d3bbb16913cd1.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2
4
2013
12
01
Graph theoretical methods to study controllability and leader selection for dead-time systems
25
36
EN
Majdeddin
Najafi
Avionics Research Institute
najafi_majd@yahoo.com
Farid
Shaikholeslam
Electrical & Computer Engineering Department
sheikh@cc.iut.ac.ir
10.22108/toc.2013.3616
In 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.
Graph Methods,Dead-time Systems,Multi-agent systems,Structural Controllability
http://toc.ui.ac.ir/article_3616.html
http://toc.ui.ac.ir/article_3616_b2a99bce0bfaf49e3d546e59ee0937c0.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2
4
2013
12
01
Graphs cospectral with a friendship graph or its complement
37
52
EN
Alireza
Abdollahi
University of Isfahan
a.abdollahi@math.ui.ac.ir
Shahrooz
Janbaz
University of Isfahan
shahrooz.janbaz@sci.ui.ac.ir
Mohammad Reza
Oboudi
University of Isfahan
mr.oboudi@sci.ui.ac.ir
10.22108/toc.2013.3621
Let $n$ 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 $Gcong F_n$. Moreover if $G$ is connected and planar then $Gcong F_n$. 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 $Hcong overline{F_n}$.
Friendship graphs,cospectral graphs,adjacency eigenvalues
http://toc.ui.ac.ir/article_3621.html
http://toc.ui.ac.ir/article_3621_0bcd0f5df9a893e748683a0325f8cac6.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2
4
2013
12
01
Directionally $n$-signed graphs-III: the notion of symmetric balance
53
62
EN
P.Siva Kota
Reddy
Dept. of Mathematics, Siddaganga Institute of Technology, B.H.Road,Tumkur-572103, India.
reddy_math@yahoo.com
U. K.
Misra
Berhampur University
10.22108/toc.2013.3658
Let $G=(V, E)$ 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 $1leq k leq n$. In this paper, we give different notion of balance: symmetric balance in a $(n,d)$-sigraph and obtain some characterizations.
Signed graphs,Directional labeling,Complementation,Balance
http://toc.ui.ac.ir/article_3658.html
http://toc.ui.ac.ir/article_3658_afa2e9837ed9f7e3782ebee733da0db9.pdf
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2
4
2013
12
01
Full friendly index sets of slender and flat cylinder graphs
63
80
EN
Wai Chee
Shiu
Hong Kong Baptist University
wcshiu@hkbu.edu.hk
Man-Ho
Ho
Hong Kong Baptist University
homanho@hkbu.edu.hk
10.22108/toc.2013.3678
Let $G=(V,E)$ 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 $P_2times P_n$, and cylinders $C_mtimes P_n$ for some $n$ and $m$. In this paper we study the full friendly index sets of cylinder graphs $C_mtimes P_2$ for $mgeq 3$, $C_mtimes P_3$ for $mgeq 4$ and $C_3times P_n$ for $ngeq 4$. The results in this paper complement the existing results in literature, so the full friendly index set of cylinder graphs are completely determined.
Full friendly index sets,friendly labeling,cylinder graphs
http://toc.ui.ac.ir/article_3678.html
http://toc.ui.ac.ir/article_3678_3225bdc21d140f967414ef14f7247734.pdf