University of IsfahanTransactions on Combinatorics2251-86575120160301Skew Randi'c matrix and skew Randi'c energy114951310.22108/toc.2016.9513ENRanGuCenter for Combinatorics, Nankai University, Tianjin 300071, P.R. ChinaFeiHuangCenter for Combinatorics, Nankai University, Tianjin 300071, P.R. ChinaXueliangLiCenter for Combinatorics, Nankai University, Tianjin 300071, ChinaJournal Article20141227Let $G$ 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.https://toc.ui.ac.ir/article_9513_5dd2d75be3009b50ce663bc68f39cb1e.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575120160301Skew equienergetic digraphs1523937210.22108/toc.2016.9372ENHarishchandra S.RamaneKarnatak University, Dharwad, India0000-0003-3122-1669K. ChannegowdaNandeeshKarnatak University, DharwadIvanGutmanUniversity of Kragujevac, 34000 Kragujevac0000-0001-9681-1550XueliangLiNankai University, TianjinJournal Article20141219Let $D$ 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]. <br /> https://toc.ui.ac.ir/article_9372_1d921f94d58d62e8f06d43db2dc426d5.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575120160301Weighted Szeged indices of some graph operations2535859410.22108/toc.2016.8594ENKannanPattabiramanAnnamalai UniversityP.KandanAnnamalai UniversityJournal Article20131202In 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.https://toc.ui.ac.ir/article_8594_7eec41f6c4504ac1095447614ce5721c.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575120160301ON $\bullet$-LICT signed graohs $L_{\bullet_c}(S)$ and $\bullet$-LINE signed graohs $L_\bullet(S)$3748789010.22108/toc.2016.7890ENMuktiAcharyaDELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIARashmiJainDELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIASangitaKansalDELHI TECHNOLOGICAL UNIVERSITY,
DELHI - INDIAJournal Article20130929A <em>signed graph</em> (or, in short, <em>sigraph</em>) $S=(S^u,\sigma)$ consists of an underlying graph $S^u :=G=(V,E)$ and a function $\sigma:E(S^u)\longrightarrow \{+,-\}$, called the signature of $S$. A <em>marking</em> of $S$ is a function $\mu:V(S)\longrightarrow \{+,-\}$. The <em>canonical marking</em> of a signed graph $S$, denoted $\mu_\sigma$, is given as $$\mu_\sigma(v) := \prod_{vw\in E(S)}\sigma(vw).$$ <br />The <em>line graph</em> 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 <em>line signed graph</em> 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 $e'$. <br /><br />The line-cut graph (or, in short, <em>lict graph</em>) 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 $c$. <br /><br />In this paper, we introduce <em>dot-lict signed graph</em> (or $\bullet$<em>-lict signed graph</em>} $L_{\bullet_c}(S)$, 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.https://toc.ui.ac.ir/article_7890_ce0590d708f808d60b29942f13824a78.pdfUniversity of IsfahanTransactions on Combinatorics2251-86575120160301Ordering of trees by multiplicative second Zagreb index4955995610.22108/toc.2016.9956ENMehdiEliasiDepartment of Mathematics and Computer Science , Faculty of Khansar, Khansar, Iran0000-0003-0721-7221AliGhalavandDepartment of
Mathematics and Computer Science, Faculty of Khansar, University
of Isfahan, P.O.Box 87931133111, Khansar, Iran0000000281860146Journal Article20150608For 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 $G$. <br />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 $n\geq 14$. <br /> https://toc.ui.ac.ir/article_9956_8efd28a3432a71695fc6a83d711c626e.pdf