Skew Randi'c matrix and skew Randi'c energy
Ran
Gu
Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China
author
Fei
Huang
Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China
author
Xueliang
Li
Center for Combinatorics, Nankai University, Tianjin 300071, China
author
text
article
2016
eng
Let $G$ be a simple graph with an orientation $\sigma$, which assigns to each edge a direction so that $G^\sigma$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^\sigma$. In this paper, we define a weighted skew adjacency matrix with Rand\'c weight, the skew Randi\'c matrix ${\bf R_S}(G^\sigma)$, of $G^\sigma$ as the real skew symmetric matrix $[(r_s)_{ij}]$ where $(r_s)_{ij} = (d_id_j)^{-\frac{1}{2}}$ and $(r_s)_{ji} = -(d_id_j)^{-\frac{1}{2}}$ if $v_i \rightarrow v_j$ is an arc of $G^\sigma$, otherwise $(r_s)_{ij} = (r_s)_{ji} = 0$. We derive some properties of the skew Randi\'c energy of an oriented graph. Most properties are similar to those for the skew energy of oriented graphs. But, surprisingly, the extremal oriented graphs with maximum or minimum skew Randi\'c energy are completely different, no longer being some kinds of oriented regular graphs.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
1
no.
2016
1
14
http://toc.ui.ac.ir/article_9513_5dd2d75be3009b50ce663bc68f39cb1e.pdf
dx.doi.org/10.22108/toc.2016.9513
Skew equienergetic digraphs
Harishchandra S.
Ramane
Karnatak University, Dharwad, India
author
K. Channegowda
Nandeesh
Karnatak University, Dharwad
author
Ivan
Gutman
University of Kragujevac, 34000 Kragujevac
author
Xueliang
Li
Nankai University, Tianjin
author
text
article
2016
eng
Let $D$ be a digraph with skew-adjacency matrix $S(D)$. The skew energy of $D$ is defined as the sum of the norms of all eigenvalues of $S(D)$. Two digraphs are said to be skew equienergetic if their skew energies are equal. We establish an expression for the characteristic polynomial of the skew adjacency matrix of the join of two digraphs, and for the respective skew energy, and thereby construct non-cospectral, skew equienergetic digraphs on $n$ vertices, for all $n \geq 6$. Thus we arrive at the solution of some open problems proposed in [X. Li, H. Lian, A survey on the skew energy of oriented graphs, arXiv:1304.5707].
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
1
no.
2016
15
23
http://toc.ui.ac.ir/article_9372_1d921f94d58d62e8f06d43db2dc426d5.pdf
dx.doi.org/10.22108/toc.2016.9372
Weighted Szeged indices of some graph operations
Kannan
Pattabiraman
Annamalai University
author
P.
Kandan
Annamalai University
author
text
article
2016
eng
In this paper, the weighted Szeged indices of Cartesian product and Corona product of two connected graphs are obtained. Using the results obtained here, the weighted Szeged indices of the hypercube of dimension $n$, Hamming graph, $C_4$ nanotubes, nanotorus, grid, $t-$fold bristled, sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
1
no.
2016
25
35
http://toc.ui.ac.ir/article_8594_7eec41f6c4504ac1095447614ce5721c.pdf
dx.doi.org/10.22108/toc.2016.8594
ON $\bullet$-LICT signed graohs $L_{\bullet_c}(S)$ and $\bullet$-LINE signed graohs $L_\bullet(S)$
Mukti
Acharya
DELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIA
author
Rashmi
Jain
DELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIA
author
Sangita
Kansal
DELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIA
author
text
article
2016
eng
A signed graph (or, in short, sigraph) $S=(S^u,\sigma)$ consists of an underlying graph $S^u :=G=(V,E)$ and a function $\sigma:E(S^u)\longrightarrow \{+,-\}$, called the signature of $S$. A marking of $S$ is a function $\mu:V(S)\longrightarrow \{+,-\}$. The canonical marking of a signed graph $S$, denoted $\mu_\sigma$, is given as $$\mu_\sigma(v) := \prod_{vw\in E(S)}\sigma(vw).$$ The line graph of a graph $G$, denoted $L(G)$, is the graph in which edges of $G$ are represented as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in $G$. There are three notions of a line signed graph of a signed graph $S=(S^u,\sigma)$ in the literature, viz., $L(S)$, $L_\times(S)$ and $L_\bullet(S)$, all of which have $L(S^u)$ as their underlying graph; only the rule to assign signs to the edges of $L(S^u)$ differ. Every edge $ee'$ in $L(S)$ is negative whenever both the adjacent edges $e$ and $e'$ in S are negative, an edge $ee'$ in $L_\times(S)$ has the product $\sigma(e)\sigma(e')$ as its sign and an edge $ee'$ in $L_\bullet(S)$ has $\mu_\sigma(v)$ as its sign, where $v\in V(S)$ is a common vertex of edges $e$ and $e'$. The line-cut graph (or, in short, lict graph) of a graph $G=(V,E)$, denoted by $L_c(G)$, is the graph with vertex set $E(G)\cup C(G)$, where $C(G)$ is the set of cut-vertices of $G$, in which two vertices are adjacent if and only if they correspond to adjacent edges of $G$ or one vertex corresponds to an edge $e$ of $G$ and the other vertex corresponds to a cut-vertex $c$ of $G$ such that $e$ is incident with $c$. In this paper, we introduce dot-lict signed graph (or $\bullet$-lict signed graph} $L_{\bullet_c}(S)$, which has $L_c(S^u)$ as its underlying graph. Every edge $uv$ in $L_{\bullet_c}(S)$ has the sign $\mu_\sigma(p)$, if $u, v \in E(S)$ and $p\in V(S)$ is a common vertex of these edges, and it has the sign $\mu_\sigma(v)$, if $u\in E(S)$ and $v\in C(S)$. we characterize signed graphs on $K_p$, $p\geq2$, on cycle $C_n$ and on $K_{m,n}$ which are $\bullet$-lict signed graphs or $\bullet$-line signed graphs, characterize signed graphs $S$ so that $L_{\bullet_c}(S)$ and $L_\bullet(S)$ are balanced. We also establish the characterization of signed graphs $S$ for which $S\sim L_{\bullet_c}(S)$, $S\sim L_\bullet(S)$, $\eta(S)\sim L_{\bullet_c}(S)$ and $\eta(S)\sim L_\bullet(S)$, here $\eta(S)$ is negation of $S$ and $\sim$ stands for switching equivalence.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
1
no.
2016
37
48
http://toc.ui.ac.ir/article_7890_ce0590d708f808d60b29942f13824a78.pdf
dx.doi.org/10.22108/toc.2016.7890
Ordering of trees by multiplicative second Zagreb index
Mehdi
Eliasi
Department of Mathematics and Computer Science , Faculty of Khansar, Khansar, Iran
author
Ali
Ghalavand
Department of
Mathematics and Computer Science, Faculty of Khansar, University
of Isfahan, P.O.Box 87931133111, Khansar, Iran
author
text
article
2016
eng
For a graph $G$ with edge set $E(G)$, the multiplicative second Zagreb index of $G$ is defined as $\Pi_2(G)=\Pi_{uv\in E(G)}[d_G(u)d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$. In this paper, we identify the eighth class of trees, with the first through eighth smallest multiplicative second Zagreb indeces among all trees of order $n\geq 14$.
Transactions on Combinatorics
University of Isfahan
2251-8657
5
v.
1
no.
2016
49
55
http://toc.ui.ac.ir/article_9956_8efd28a3432a71695fc6a83d711c626e.pdf
dx.doi.org/10.22108/toc.2016.9956