University of Isfahan Transactions on Combinatorics 2251-8657 5 1 2016 03 01 Skew Randi'c matrix and skew Randi'c energy 1 14 9513 10.22108/toc.2016.9513 EN Ran Gu Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China Fei Huang Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China Xueliang Li Center for Combinatorics, Nankai University, Tianjin 300071, China Journal Article 2014 12 27 ‎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 <br />‎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 <br />‎different‎, ‎no longer being some kinds of oriented regular graphs‎. http://toc.ui.ac.ir/article_9513_5dd2d75be3009b50ce663bc68f39cb1e.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 5 1 2016 03 01 Skew equienergetic digraphs 15 23 9372 10.22108/toc.2016.9372 EN Harishchandra S. Ramane Karnatak University, Dharwad, India K. Channegowda Nandeesh Karnatak University, Dharwad Ivan Gutman University of Kragujevac, 34000 Kragujevac Xueliang Li Nankai University, Tianjin Journal Article 2014 12 19 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]‎. <br />  http://toc.ui.ac.ir/article_9372_1d921f94d58d62e8f06d43db2dc426d5.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 5 1 2016 03 01 Weighted Szeged indices of some graph operations 25 35 8594 10.22108/toc.2016.8594 EN Kannan Pattabiraman Annamalai University P. Kandan Annamalai University Journal Article 2013 12 02 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. http://toc.ui.ac.ir/article_8594_7eec41f6c4504ac1095447614ce5721c.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 5 1 2016 03 01 ON \$bullet\$-LICT signed graohs \$L_{bullet_c}(S)\$ and \$bullet\$-LINE signed graohs \$L_bullet(S)\$ 37 48 7890 10.22108/toc.2016.7890 EN Mukti Acharya DELHI TECHNOLOGICAL UNIVERSITY, DELHI - INDIA Rashmi Jain DELHI TECHNOLOGICAL UNIVERSITY, DELHI - INDIA Sangita Kansal DELHI TECHNOLOGICAL UNIVERSITY, DELHI - INDIA Journal Article 2013 09 29 A <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_{vwin 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 \$vin 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 \$pin V(S)\$ is a common vertex of these edges‎, ‎and it has the sign \$mu_sigma(v)\$‎, ‎if \$uin E(S)\$ and \$vin C(S)\$‎. ‎we characterize signed graphs on \$K_p\$‎, ‎\$pgeq2\$‎, ‎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 \$Ssim L_{bullet_c}(S)\$‎, ‎\$Ssim 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. http://toc.ui.ac.ir/article_7890_ce0590d708f808d60b29942f13824a78.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 5 1 2016 03 01 Ordering of trees by multiplicative second Zagreb index 49 55 9956 10.22108/toc.2016.9956 EN Mehdi Eliasi Department of Mathematics and Computer Science , Faculty of Khansar, Khansar, Iran Ali Ghalavand Department of Mathematics and Computer Science, Faculty of Khansar, University of Isfahan, P.O.Box 87931133111, Khansar, Iran Journal Article 2015 06 08 ‎For a graph \$G\$ with edge set \$E(G)\$‎, ‎the multiplicative second Zagreb index of \$G\$ is defined as‎ ‎\$Pi_2(G)=Pi_{uvin 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 \$ngeq 14\$‎. <br />  http://toc.ui.ac.ir/article_9956_8efd28a3432a71695fc6a83d711c626e.pdf