Transactions on Combinatorics
Volume:5 Issue: 1, Mar 2016

Let G G be a simple graph with an orientation σ σ ‎, ‎which ‎assigns to each edge a direction so that G σ Gσ becomes a‎ ‎directed graph‎. ‎G G is said to be the underlying graph of the‎ ‎directed graph G σ Gσ ‎. ‎In this paper‎, ‎we define a weighted skew‎ ‎adjacency matrix with Randc weight‎, ‎the skew Randic matrix ‎‎R S (G σ) ‎‎RS(Gσ) ‎, ‎of G σ Gσ as the real skew symmetric matrix‎ ‎[(r s) ij] [(rs)ij] where (r s) ij =(d i d j) −12 (rs)ij=(didj)−12 and‎ ‎(r s) ji =‎−‎(d i d j) −12 (rs)ji=‎−‎(didj)−12 if v i →v j vi→vj is‎ ‎an arc of G σ Gσ ‎, ‎otherwise (r s) ij =(r s) ji =0 (rs)ij=(rs)ji=0 ‎. ‎We‎ ‎derive some properties of the skew Randic 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 Randic energy are completely‎ ‎different‎, ‎no longer being some kinds of oriented regular graphs‎.

Let D D be a digraph with skew-adjacency matrix S(D) S(D) ‎. ‎The skew‎ ‎energy of D D is defined as the sum of the norms of all‎ ‎eigenvalues of S(D) 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 n vertices‎, ‎for all n≥6 n≥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]‎.

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, C4 nanotubes, nanotorus, grid, t− foldbristled, sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.

A emph{signed graph} (or, in short, emph{sigraph}) S=(S u, sigma) S=(Su,sigma) consists of an underlying graph S u: =G=(V,E) Su:=G=(V,E) and a function sigma:E(S u)longrightarrow+,− sigma:E(Su)longrightarrow+,−, called the signature of S S. A emph{marking} of S S is a function mu:V(S)longrightarrow+,− mu:V(S)longrightarrow+,−. The emph{canonical marking} of a signed graph S S, denoted mu s igma musigma, is given as mu s igma(v):=prod vwinE(S) sigma(vw). musigma(v):=prodvwinE(S)sigma(vw). The line-cut graph (or, in short, emph{lict graph}) of a graph G=(V,E) G=(V,E), denoted by L c (G) Lc(G), is the graph with vertex set E(G)cupC(G) E(G)cupC(G), where C(G) C(G) is the set of cut-vertices of G G, in which two vertices are adjacent if and only if they correspond to adjacent edges of G G or one vertex corresponds to an edge e e of G G and the other vertex corresponds to a cut-vertex c c of G G such that e e is incident with c c. In this paper, we introduce emph{Dot-lict signed graph} (or emph{bullet bullet -lict signed graph}) L bullet c (S) Lbulletc(S), which has L c (S u) Lc(Su) as its underlying graph. Every edge uv uv in L bullet c (S) Lbulletc(S) has the sign mu s igma(p) musigma(p), if u,vinE(S) u,vinE(S) and pinV(S) pinV(S) is a common vertex of these edges, and it has the sign mu s igma(v) musigma(v), if uinE(S) uinE(S) and vinC(S) vinC(S). we characterize signed graphs on K p Kp, pgeq2 pgeq2, on cycle C n Cn and on K m,n Km,n which are bullet bullet -lict signed graphs or bullet bullet -line signed graphs, characterize signed graphs S S so that L bullet c (S) Lbulletc(S) and L b ullet(S) Lbullet(S) are balanced. We also establish the characterization of signed graphs S S for which SsimL bullet c (S) SsimLbulletc(S), SsimL b ullet(S) SsimLbullet(S), eta(S)simL bullet c (S) eta(S)simLbulletc(S) and eta(S)simL b ullet(S) eta(S)simLbullet(S), here eta(S) eta(S) is negation of S S and sim sim stands for switching equivalence.

‎For a graph G G with edge set E(G) E(G) ‎, ‎the multiplicative second Zagreb index of G G is defined as‎ ‎Π 2 (G)=Π uv∈E(G) [d G (u)d G (v)] Π2(G)=Πuv∈E(G)[dG(u)dG(v)] ‎, ‎where d G (v) dG(v) is the degree of vertex v v in G 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≥14 n≥14 ‎.