Peacock Graphs are Determined by Their Laplacian Spectra

A peacock graph PG = (i, j, k, b(1), b(2) ,..., b(s)), where i, j, k >= 3 is a graph consisting of three cycles C-i, C-j, C-k and so (>= 1) paths P-b1+1, P-b2+1,..., P-bs+1 intersecting in a single vertex that all meet in one vertex. A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS, for short) if every graph with the same Laplacian spectrum is isomorphic to G. If s = 1, then the peacock graph is called clover graph. In Wang and Wang (Linear Multilinear Algebra 63(12):2396-2405, 2015), it was proved that all clover graphs are DLS. In this paper, we generalize this result and show that all peacock graphs are DLS.