ORIGINAL_ARTICLE
Randic incidence energy of graphs
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.
http://toc.ui.ac.ir/article_5573_68f2261c2087d1f09fb34c2f8de4b053.pdf
2014-12-01T11:23:20
2017-10-23T11:23:20
1
9
Randi'c incidence matrix
Randi'c incidence energy
eigenvalues
Ran
Gu
guran323@163.com
true
1
Nankai University
Nankai University
Nankai University
AUTHOR
Fei
Huang
huangfei06@126.com
true
2
Nankai University
Nankai University
Nankai University
AUTHOR
Xueliang
Li
lxl@nankai.edu.cn
true
3
Center for Combinatorics, Nankai University, Tianjin 300071, China
Center for Combinatorics, Nankai University, Tianjin 300071, China
Center for Combinatorics, Nankai University, Tianjin 300071, China
LEAD_AUTHOR
1] J. A. Bondy and U. S. R. Murty, Graph Theory, 244, Springer, New York, 2008.
1
[2] S . B. Bozkurt and I. Gutman, Estimating the incidence energy, MATCH Commun. Math. Comput. Chem., 70 (2013) 143–156.
2
[3] G. Caporossi, I. Gutman, P. Hansen and L. Pavlovic, Graphs with maximum connectivity index, Comput. Biol. Chem., 27 (2003) 85–90.
3
[4] F. R. K. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence, 1997.
4
[5] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs-Theory and Application, Third ed., Johann Ambrosius Barth. Heidelberg, 1995.
5
[6] D. Cvetkovic, P. Rowlinson and S. Simic, Signless Laplacians of finite graphs, Linear Algebra Appl., 423 (2007) 155–171.
6
[7] K. C. Das and I. Gutman, On incidence energy of graphs, Linear Algebra Appl., 446 (2014) 329–344.
7
[8] R. Gu, F. Huang and X. Li, General Randi´ c matrix and general Randic energy, Trans. Comb., 3 no. 3 (2014) 21-33.
8
[9] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz, 103 (1978) 1–22.
9
[10] I. Gutman, B. Furtula and S . B. Bozkurt, On Randic energy, Linear Algebra Appl., 442 (2014) 50–57.
10
[11] I. Gutman, D. Kiani and M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem., 62 (2009) 573–580.
11
[12] I. Gutman, D. Kiani, M. Mirzakhah and B. Zhou, On incidence energy of a graph, Linear Algebra Appl., 431 (2009) 1223–1233.
12
[13] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
13
[14] M. Jooyandeh, D. Kiani and M. Mirzakhah, Incidence energy of a graph, MATCH Commun. Math. Comput. Chem., 62 (2009) 561–572.
14
[15] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.
15
[16] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl., 326 (2007) 1472–1475.
16
[17] M. Randic, On characterization of molecular branching, J. Amer. Chem. Soc., 97 (1975) 6609–6615.
17
[18] P. Yu, An upper bound on the Randi´ c index of trees, (Chinese), J. Math. Study, 31 (1998) 225–230.
18
ORIGINAL_ARTICLE
On Lict sigraphs
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.
http://toc.ui.ac.ir/article_5627_e7de2aef7c26e21d97bfaf79f2112406.pdf
2014-12-01T11:23:20
2017-10-23T11:23:20
11
18
Signed graph
Line sigraph
Jump sigraph
Semitotal line sigraph
Lict sigraph
Veena
Mathad
veena_mathad@rediffmail.com
true
1
University of Mysore
University of Mysore
University of Mysore
LEAD_AUTHOR
Kishori
Narayankar
kishori_pn@yahoo.co.in
true
2
Mangalore University
Mangalore University
Mangalore University
AUTHOR
ORIGINAL_ARTICLE
The geodetic domination number for the product of graphs
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
2014-12-01T11:23:20
2017-10-23T11:23:20
19
30
Cartesian product
strong product
geodetic number
Domination Number
geodetic domination number
S. Robinson
Chellathurai
robinchel@rediffmail.com
true
1
Scott Christian College
Scott Christian College
Scott Christian College
AUTHOR
S. Padma
Vijaya
padmaberry@yahoo.com
true
2
University College of Engineering Nagercoil
University College of Engineering Nagercoil
University College of Engineering Nagercoil
LEAD_AUTHOR
ORIGINAL_ARTICLE
Comparing the second multiplicative Zagreb coindex with some graph invariants
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.
http://toc.ui.ac.ir/article_5951_503474fbf2c1d8a206d3e6dd20f8a32e.pdf
2014-12-01T11:23:20
2017-10-23T11:23:20
31
41
Degree (in graphs)
Topological index
multiplicative Zagreb coindex
Farzaneh
Falahati Nezhad
farzanehfalahati_n@yahoo.com
true
1
Science and Research Branch, Islamic Azad University
Science and Research Branch, Islamic Azad University
Science and Research Branch, Islamic Azad University
AUTHOR
Ali
Iranmanesh
iranmanesh@modares.ac.ir
true
2
Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran
Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran
Department of Mathematics, Tarbiat Modares University, P. O. Box 14115-137, Tehran
LEAD_AUTHOR
Abolfazl
Tehranian
tehranian@srbiau.ac.ir
true
3
Science and Research Branch, Islamic Azad University
Science and Research Branch, Islamic Azad University
Science and Research Branch, Islamic Azad University
AUTHOR
Mahdieh
Azari
azari@kau.ac.ir
true
4
Kazerun Branch, Islamic Azad University
Kazerun Branch, Islamic Azad University
Kazerun Branch, Islamic Azad University
AUTHOR
ORIGINAL_ARTICLE
Perfect state transfer in unitary Cayley graphs over local rings
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
2014-12-01T11:23:20
2017-10-23T11:23:20
43
54
Local rings
Perfect state transfer, Unitary Cayley graphs
Yotsanan
Meemark
yzm101@yahoo.com
true
1
Chulalongkorn University
Chulalongkorn University
Chulalongkorn University
LEAD_AUTHOR
Songpon
Sriwongsa
songpon_sriwongsa@hotmail.com
true
2
Chulalongkorn University
Chulalongkorn University
Chulalongkorn University
AUTHOR
ORIGINAL_ARTICLE
Complete solution to a conjecture of Zhang-Liu-Zhou
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
2014-12-01T11:23:20
2017-10-23T11:23:20
55
58
Eccentric connectivity index
tricyclic graph
tetracyclic graph
graph operation
Mostafa
Tavakoli
m_tavakoli@ferdowsi.um.ac.ir
true
1
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
LEAD_AUTHOR
F.
Rahbarnia
rahbarnia@um.ac.ir
true
2
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
AUTHOR
M.
Mirzavaziri
mirzavaziri@gmail.com
true
3
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
AUTHOR
A. R.
Ashrafi
ashrafi@kashanu.ac.ir
true
4
University of Kashan
University of Kashan
University of Kashan
AUTHOR