Spectral distances of graphs

Let λ1(G)⩾λ2(G)⩾⋯⩾λn(G) be the adjacency spectrum of a graph G on n vertices. The spectral distance σ(G1,G2) between n vertex graphs G1 and G2 is defined byσ(G1,G2)=∑i=1n|λi(G1)-λi(G2)|.Here we provide some initial results regarding this quantity. First, we give some general results concerning the spectral distances between arbitrary graphs, and compute these distances in some particular cases. Certain relation with the theory of graph energy is identified. The spectral distances bounded by a given constant are also considered. Next, we introduce the cospectrality measure and the spectral diameter, and obtain specific results indicating their relevance for the theory of cospectral graphs. Finally, we give and discuss some computational results and conclude the paper by a list of conjectures.