Polynomial relations between matrices of graphs. Issue 3 (24th September 2018) - Spiro, Sam
Abstract: We derive a correspondence between the eigenvalues of the adjacency matrix A and the signless Laplacian matrix Q of a graph G when G is ( d 1, d 2 ) ‐biregular by using the relation A 2 = ( Q − d 1 I ) ( Q − d 2 I ) . This motivates asking when it is possible to have X r = f ( Y ) for f a polynomial, r > 0, and X, Y matrices associated to a graph G . It turns out that, essentially, this can only happen if G is either regular or biregular.