Roman game domination subdivision number of a graph
Jafar
Amjadi
Azarbaijan Shahid Madani University
author
Hossein
Karami
Azarbaijan Shahid Madani University
author
Seyed Mahmoud
Sheikholeslami
Azarbaijan University of Tarbiat Moallem
author
Lutz
Volkmann
RWTH-Aachen University
author
text
article
2013
eng
A Roman dominating function 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 weight 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
2
v.
4
no.
2013
1
12
http://toc.ui.ac.ir/article_3341_b03cc8118595dcee034dcf7f43bede8d.pdf
dx.doi.org/10.22108/toc.2013.3341
Reciprocal degree distance of some graph operations
Kannan
Pattabiraman
Annamalai University
author
M.
Vijayaragavan
Thiruvalluvar College of Engineering and Technology
author
text
article
2013
eng
The 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
2
v.
4
no.
2013
13
24
http://toc.ui.ac.ir/article_3506_bfc02f2003611e70f89d3bbb16913cd1.pdf
dx.doi.org/10.22108/toc.2013.3506
Graph theoretical methods to study controllability and leader selection for dead-time systems
Majdeddin
Najafi
Avionics Research Institute
author
Farid
Shaikholeslam
Electrical & Computer Engineering Department
author
text
article
2013
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
2
v.
4
no.
2013
25
36
http://toc.ui.ac.ir/article_3616_b2a99bce0bfaf49e3d546e59ee0937c0.pdf
dx.doi.org/10.22108/toc.2013.3616
Graphs cospectral with a friendship graph or its complement
Alireza
Abdollahi
University of Isfahan
author
Shahrooz
Janbaz
University of Isfahan
author
Mohammad Reza
Oboudi
University of Isfahan
author
text
article
2013
eng
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 $G\cong F_n$. 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 $H\cong \overline{F_n}$.
Transactions on Combinatorics
University of Isfahan
2251-8657
2
v.
4
no.
2013
37
52
http://toc.ui.ac.ir/article_3621_0bcd0f5df9a893e748683a0325f8cac6.pdf
dx.doi.org/10.22108/toc.2013.3621
Directionally $n$-signed graphs-III: the notion of symmetric balance
P.Siva Kota
Reddy
Dept. of Mathematics, Siddaganga Institute of Technology, B.H.Road,Tumkur-572103, India.
author
U. K.
Misra
Berhampur University
author
text
article
2013
eng
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 $1\leq k \leq n$. In this paper, we give different notion of balance: symmetric balance in a $(n,d)$-sigraph and obtain some characterizations.
Transactions on Combinatorics
University of Isfahan
2251-8657
2
v.
4
no.
2013
53
62
http://toc.ui.ac.ir/article_3658_afa2e9837ed9f7e3782ebee733da0db9.pdf
dx.doi.org/10.22108/toc.2013.3658
Full friendly index sets of slender and flat cylinder graphs
Wai Chee
Shiu
Hong Kong Baptist University
author
Man-Ho
Ho
Hong Kong Baptist University
author
text
article
2013
eng
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 $P_2\times P_n$, 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 $m\geq 4$ and $C_3\times P_n$ for $n\geq 4$. The results in this paper complement the existing results in literature, so the full friendly index set of cylinder graphs are completely determined.
Transactions on Combinatorics
University of Isfahan
2251-8657
2
v.
4
no.
2013
63
80
http://toc.ui.ac.ir/article_3678_3225bdc21d140f967414ef14f7247734.pdf
dx.doi.org/10.22108/toc.2013.3678