GRAPHS THAT ARE COSPECTRAL FOR THE DISTANCE LAPLACIAN

The distance matrix D(G) of a graph G is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is D-L(G) = T(G) - D(G), where T(G) is the diagonal matrix of row sums of D(G). Several general methods are established for producing D-L-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by D-L-cospectrality, including examples of D-L-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., vertical bar delta(L)(1)vertical bar >= ... >= vertical bar delta(L)(n)vertical bar, where delta(L)(k) is the coefficient of x(k).