A note on the automorphism groups of cubic Cayley graphs of finite simple groups

For a finite group G, a Cayley graph on G is said to be normal if . In this note, we prove that connected cubic non-symmetric Cayley graphs of the ten finite non-abelian simple groups G in the list of non-normal candidates given in [X.G. Fang, C.H. Li, J. Wang, M.Y. Xu, On cubic Cayley graphs of finite simple groups, Discrete Math. 244 (2002) 67-75] are normal.     

A class of finite groups with complete character degree graphs

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

The Laplacian spectral radius of graphs on surfaces

Let G be an n-vertex (n?3) simple graph embeddable on a surface of Euler genus γ (the number of crosscaps plus twice the number of handles). Denote by Δ the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows:

(i)

     

On the independence number of the power graph of a finite group

The power graph ${\Gamma }_G$ of a finite group $G$ is the graph whose vertex set is $G$, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ${\Gamma }_G$ and characterize the groups achieving the bounds. Moreover, we determine the independence number of ${\Gamma }_G$ if $G$ is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups $G$ whose power graphs have independence number 3 or $n?2$, where $n$ is the order of $G$.     

A sharp upper bound on the spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained.     

The new upper bounds on the spectral radius of weighted graphs

Let us consider weighted graphs, where the weights of the edges are positive definite matrices. The eigenvalues of a weighted graph are the eigenvalues of its adjacency matrix and the spectral radius of a weighted graph is also the spectral radius of its adjacency matrix. In this paper, we obtain two upper bounds for the spectral radius of weighted graphs and compare with a known upper bound. We also characterize graphs for which the upper bounds are attained.