eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2014-12-01
3
4
1
9
10.22108/toc.2014.5573
5573
Randic incidence energy of graphs
Ran Gu
guran323@163.com
1
Fei Huang
huangfei06@126.com
2
Xueliang Li
lxl@nankai.edu.cn
3
Nankai University
Nankai University
Center for Combinatorics, Nankai University, Tianjin 300071, China
Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots, v_n}$ and edge set $E(G) = {e_1, e_2,ldots, e_m}$. Similar to the Randi'c matrix, here we introduce the Randi'c incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $ntimes m$ matrix whose $(i,j)$-entry is $(d_i)^{-frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randi'c incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randic incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all $n$-vertex trees. As a result, some results are very different from those for incidence energy.
http://toc.ui.ac.ir/article_5573_68f2261c2087d1f09fb34c2f8de4b053.pdf
Randi'c incidence matrix
Randi'c incidence energy
eigenvalues
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2014-12-01
3
4
11
18
10.22108/toc.2014.5627
5627
On Lict sigraphs
Veena Mathad
veena_mathad@rediffmail.com
1
Kishori Narayankar
kishori_pn@yahoo.co.in
2
University of Mysore
Mangalore University
A signed graph (marked graph) is an ordered pair $S=(G,sigma)$ $(S=(G,mu))$, where $G=(V,E)$ is a graph called the underlying graph of $S$ and $sigma:Erightarrow{+,-}$ $(mu:Vrightarrow{+,-})$ is a function. For a graph $G$, $V(G), E(G)$ and $C(G)$ denote its vertex set, edge set and cut-vertex set, respectively. The lict graph $L_{c}(G)$ of a graph $G=(V,E)$ is defined as the graph having vertex set $E(G)cup C(G)$ in which two vertices are adjacent if and only if they correspond to adjacent edges of $G$ or one corresponds to an edge $e_{i}$ of $G$ and the other corresponds to a cut-vertex $c_{j}$ of $G$ such that $e_{i}$ is incident with $c_{j}$. In this paper, we introduce lict sigraphs, as a natural extension of the notion of lict graph to the realm of signed graphs. We show that every lict sigraph is balanced. We characterize signed graphs $S$ and $S^{'}$ for which $Ssim L_{c}(S)$, $eta(S)sim L_{c}(S)$, $L(S)sim L_{c}(S')$, $J(S)sim L_{c}(S^{'})$ and $T_{1}(S)sim L_{c}(S^{'})$, where $eta(S)$, $L(S)$, $J(S)$ and $T_{1}(S)$ are negation, line graph, jump graph and semitotal line sigraph of $S$, respectively, and $sim$ means switching equivalence.
http://toc.ui.ac.ir/article_5627_e7de2aef7c26e21d97bfaf79f2112406.pdf
signed graph
Line sigraph
Jump sigraph
Semitotal line sigraph
Lict sigraph
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2014-12-01
3
4
19
30
10.22108/toc.2014.5750
5750
The geodetic domination number for the product of graphs
S. Robinson Chellathurai
robinchel@rediffmail.com
1
S. Padma Vijaya
padmaberry@yahoo.com
2
Scott Christian College
University College of Engineering Nagercoil
A subset $S$ of vertices in a graph $G$ is called a geodetic set if every vertex not in $S$ lies on a shortest path between two vertices from $S$. A subset $D$ of vertices in $G$ is called dominating set if every vertex not in $D$ has at least one neighbor in $D$. A geodetic dominating set $S$ is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number $g(G) (gamma(G),gamma_g(G))$ of $G$ is the minimum cardinality among all geodetic (dominating, geodetic dominating) sets in $G$. In this paper, we show that if a triangle free graph $G$ has minimum degree at least 2 and $g(G) = 2$, then $gamma _g(G) = gamma(G)$. It is shown, for every nontrivial connected graph $G$ with $gamma(G) = 2$ and $diam(G) > 3$, that $gamma_g(G) > g(G)$. The lower bound for the geodetic domination number of Cartesian product graphs is proved. Geodetic domination number of product of cycles (paths) are determined. In this work, we also determine some bounds and exact values of the geodetic domination number of strong product of graphs.
http://toc.ui.ac.ir/article_5750_87bdcf395fa6e3fd7e39a154bc0f1442.pdf
Cartesian product
strong product
geodetic number
Domination Number
geodetic domination number
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2014-12-01
3
4
31
41
10.22108/toc.2014.5951
5951
Comparing the second multiplicative Zagreb coindex with some graph invariants
Farzaneh Falahati Nezhad
farzanehfalahati_n@yahoo.com
1
Ali Iranmanesh
iranmanesh@modares.ac.ir
2
Abolfazl Tehranian
tehranian@srbiau.ac.ir
3
Mahdieh Azari
azari@kau.ac.ir
4
Science and Research Branch, Islamic Azad University
Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran
Science and Research Branch, Islamic Azad University
Kazerun Branch, Islamic Azad University
The second multiplicative Zagreb coindex of a simple graph $G$ is defined as: $${overline{Pi{}}}_2left(Gright)=prod_{uvnotin{}E(G)}d_Gleft(uright)d_Gleft(vright),$$ where $d_Gleft(uright)$ denotes the degree of the vertex $u$ of $G$. In this paper, we compare $overline{{Pi}}_2$-index with some well-known graph invariants such as the Wiener index, Schultz index, eccentric connectivity index, total eccentricity, eccentric-distance sum, the first Zagreb index and coindex and the first multiplicative Zagreb index and coindex.
http://toc.ui.ac.ir/article_5951_503474fbf2c1d8a206d3e6dd20f8a32e.pdf
Degree (in graphs)
Topological index
multiplicative Zagreb coindex
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2014-12-01
3
4
43
54
10.22108/toc.2014.5974
5974
Perfect state transfer in unitary Cayley graphs over local rings
Yotsanan Meemark
yzm101@yahoo.com
1
Songpon Sriwongsa
songpon_sriwongsa@hotmail.com
2
Chulalongkorn University
Chulalongkorn University
In this work, using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allowing PST occurring in its unitary Cayley graph. Moreover, we have some developments when $R$ is a product of local rings.
http://toc.ui.ac.ir/article_5974_ee7986b6514f7db7db00ebf91f7de927.pdf
Local rings
Perfect state transfer, Unitary Cayley graphs
eng
University of Isfahan
Transactions on Combinatorics
2251-8657
2251-8665
2014-12-01
3
4
55
58
10.22108/toc.2014.5986
5986
Complete solution to a conjecture of Zhang-Liu-Zhou
Mostafa Tavakoli
m_tavakoli@ferdowsi.um.ac.ir
1
F. Rahbarnia
rahbarnia@um.ac.ir
2
M. Mirzavaziri
mirzavaziri@gmail.com
3
A. R. Ashrafi
ashrafi@kashanu.ac.ir
4
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
University of Kashan
Let $d_{n,m}=big[frac{2n+1-sqrt{17+8(m-n)}}{2}big]$ and $E_{n,m}$ be the graph obtained from a path $P_{d_{n,m}+1}=v_0v_1 cdots v_{d_{n,m}}$ by joining each vertex of $K_{n-d_{n,m}-1}$ to $v_{d_{n,m}}$ and $v_{d_{n,m}-1}$, and by joining $m-n+1-{n-d_{n,m}choose 2}$ vertices of $K_{n-d_{n,m}-1}$ to $v_{d_{n,m}-2}$. Zhang, Liu and Zhou [On the maximal eccentric connectivity indices of graphs, Appl. Math. J. Chinese Univ., in press] conjectured that if $d_{n,m}geqslant 3$, then $E_{n,m}$ is the graph with maximal eccentric connectivity index among all connected graph with $n$ vertices and $m$ edges. In this note, we prove this conjecture. Moreover, we present the graph with maximal eccentric connectivity index among the connected graphs with $n$ vertices. Finally, the minimum of this graph invariant in the classes of tricyclic and tetracyclic graphs are computed.
http://toc.ui.ac.ir/article_5986_740f215bc6659e95ccaa77c44e50e504.pdf
Eccentric connectivity index
tricyclic graph
tetracyclic graph
graph operation