Abstract
Let On(Q) and Sn(Z) be the set of all n by n orthogonal matrices with rational entries and the set of all n by n symmetric matrices with integer entries, respectively. We consider the following question: Given A∈Sn(Z). When does QTAQ∈Sn(Z) with Q∈On(Q) imply that Q is a signed permutation matrix? Wang and Yu [15] show that the answer to the above question is yes whenever the discriminant Δ(A) of the characteristic polynomial of A is odd and square-free. However, the above result cannot be applied for the adjacency matrix of a simple graph, since Δ(A) can never be odd in this situation. We give a new condition under which the answer is still yes for the adjacency matrices of graphs. The result was achieved by employing some tools from our recent work on the generalized spectral characterizations of graphs, which is somewhat unexpected. Some applications of the above result are also provided.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
