MONSTER GRAPHS ARE DETERMINED BY THEIR LAPLACIAN SPECTRA

A graph G is determined by its Laplacian spectrum (DLS) if every graph with the same Laplacian spectrum is isomorphic to G. A multi-fan graph is a graph of the form (P-n1 boolean OR P-n2 boolean OR center dot center dot center dot boolean OR P-nk) del K-1, where K-1 denotes the complete graph of size 1, P-n1 boolean OR P-n2 boolean OR center dot center dot center dot boolean OR P-nk is the disjoint union of paths P-ni, n(i) >= 1 and 1 <= i <= k; and a starlike tree is a tree with exactly one vertex of degree greater than 2. If a multi-fan graph and a starlike tree are joined by identifying their vertices of degree more than 2, then the resulting graph is called a monster graph. In some earlier works, it was shown that all multi-fan and path-friendship graphs are DLS. The aim of this paper is to generalize these facts by proving that all monster graphs are DLS.