Smith Normal Form and the generalized spectral characterization of oriented graphs

Abstract

Spectral characterization of graphs is an important topic in spectral graph theory. An oriented graph Gσ is obtained from a simple undirected graph G by assigning to every edge of G a direction so that Gσ becomes a directed graph. The skew-adjacency matrix of an oriented graph Gσ is a real skew-symmetric matrix S(Gσ)=(sij), where sij=−sji=1 if (i,j) is an arc; sij=sji=0 otherwise. Let Gσ and Hτ be two oriented graphs whose skew-adjacency matrices are S(Gσ) and S(Hτ), respectively. We say Gσ is R-cospectral to Hτ if tJ−S(Gσ) and tJ−S(Hτ) have the same spectrum for any t∈R, where J is the all-ones matrix. An oriented graph Gσ is said to be determined by the generalized skew spectrum (DGSS for short), if any oriented graph which is R-cospectral to Gσ is isomorphic to Gσ. Let W(Gσ)=[e,S(Gσ)e,S2(Gσ)e,…,Sn−1(Gσ)e] be the skew-walk-matrix of Gσ, where e is the all-ones vector. A theorem of Qiu, Wang and Wang [9] states that if Gσ is a self-converse oriented graph and 2−⌊n2⌋det⁡W(Gσ) is odd and square-free, then Gσ is DGSS. In this paper, based on the Smith Normal Form of the skew-walk-matrix of Gσ we obtain our main result: Let q be a prime and Gσ be a self-converse oriented graph on n vertices with det⁡W(Gσ)≠0. Assume that rankq(W(Gσ))=n−1 if q is an odd prime, and rankq(W(Gσ))=⌈n2⌉ if q=2. If Q is a regular rational orthogonal matrix satisfying QTS(Gσ)Q=S(Hτ) for some oriented graph Hτ, then the level of Q divides dn(W(Gσ))q, where dn(W(Gσ)) is the n-th invariant factor of W(Gσ). Consequently, it leads to an easier way to prove Qiu, Wang and Wang's theorem above.