Randic incidence energy of graphs
Ran
Gu
Nankai University
author
Fei
Huang
Nankai University
author
Xueliang
Li
Center for Combinatorics, Nankai University, Tianjin 300071, China
author
text
article
2014
eng
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 $n\times m$ 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
3
v.
4
no.
2014
1
9
http://toc.ui.ac.ir/article_5573_68f2261c2087d1f09fb34c2f8de4b053.pdf
dx.doi.org/10.22108/toc.2014.5573
On Lict sigraphs
Veena
Mathad
University of Mysore
author
Kishori
Narayankar
Mangalore University
author
text
article
2014
eng
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 $\sigma:E\rightarrow\{+,-\}$ $(\mu:V\rightarrow\{+,-\})$ 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 $S\sim L_{c}(S)$, $\eta(S)\sim L_{c}(S)$, $L(S)\sim L_{c}(S')$, $J(S)\sim L_{c}(S^{'})$ 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
3
v.
4
no.
2014
11
18
http://toc.ui.ac.ir/article_5627_e7de2aef7c26e21d97bfaf79f2112406.pdf
dx.doi.org/10.22108/toc.2014.5627
The geodetic domination number for the product of graphs
S. Robinson
Chellathurai
Scott Christian College
author
S. Padma
Vijaya
University College of Engineering Nagercoil
author
text
article
2014
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
3
v.
4
no.
2014
19
30
http://toc.ui.ac.ir/article_5750_87bdcf395fa6e3fd7e39a154bc0f1442.pdf
dx.doi.org/10.22108/toc.2014.5750
Comparing the second multiplicative Zagreb coindex with some graph invariants
Farzaneh
Falahati Nezhad
Science and Research Branch, Islamic Azad University
author
Ali
Iranmanesh
Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran
author
Abolfazl
Tehranian
Science and Research Branch, Islamic Azad University
author
Mahdieh
Azari
Kazerun Branch, Islamic Azad University
author
text
article
2014
eng
The second multiplicative Zagreb coindex of a simple graph $G$ is defined as: $${\overline{\Pi{}}}_2\left(G\right)=\prod_{uv\not\in{}E(G)}d_G\left(u\right)d_G\left(v\right),$$ where $d_G\left(u\right)$ 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.
Transactions on Combinatorics
University of Isfahan
2251-8657
3
v.
4
no.
2014
31
41
http://toc.ui.ac.ir/article_5951_503474fbf2c1d8a206d3e6dd20f8a32e.pdf
dx.doi.org/10.22108/toc.2014.5951
Perfect state transfer in unitary Cayley graphs over local rings
Yotsanan
Meemark
Chulalongkorn University
author
Songpon
Sriwongsa
Chulalongkorn University
author
text
article
2014
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
3
v.
4
no.
2014
43
54
http://toc.ui.ac.ir/article_5974_ee7986b6514f7db7db00ebf91f7de927.pdf
dx.doi.org/10.22108/toc.2014.5974
Complete solution to a conjecture of Zhang-Liu-Zhou
Mostafa
Tavakoli
Ferdowsi University of Mashhad
author
F.
Rahbarnia
Ferdowsi University of Mashhad
author
M.
Mirzavaziri
Ferdowsi University of Mashhad
author
A. R.
Ashrafi
University of Kashan
author
text
article
2014
eng
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.
Transactions on Combinatorics
University of Isfahan
2251-8657
3
v.
4
no.
2014
55
58
http://toc.ui.ac.ir/article_5986_740f215bc6659e95ccaa77c44e50e504.pdf
dx.doi.org/10.22108/toc.2014.5986