Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

Let $G$ be a finite group. Define graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to adjacent if and only $\langle x,y\rangle$ is cyclic. Denote this $\Delta(G)$. The $\Delta(G)$ has appeared in literature under names cyclic deleted enhanced power graph. If $H$ are nontrivial groups, then $\Delta(G\times H)$ completely characterized. In particular,...

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...

In this note, we show how to obtain a “characteristic power series” of graphons – infinite limits dense graphs as the limit normalized reciprocal characteristic polynomials. This leads new characterization graph quasi-randomness and another perspective on spectral theory for graphons, complete description function in terms spectrum graphon self-adjoint kernel operator. Interestingly, while appl...

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

Abstract In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research. AMS Classification: ...