The Laplacian spectrum of a graph
SIAM Journal on Matrix Analysis and Applications
No starlike trees are cospectral
Discrete Mathematics
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A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike trees G and H are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, let G be a simple graph of order n with vertex set V(G) = {1,2 ....,n} and let H = {H1,H2,...,Hn} be a family of rooted graphs. According to [2], the rooted product G(H) is the graph obtained by identifying the root of Hi with the i-th vertex of G. In particular, if H is the family of the paths Pk1, Pk2, ..., Pkn with the rooted vertices of degree one, in this paper the corresponding graph G(H) is called the sunlike graph and is denoted by G(k1,k2,...,kn). For any (x1,x2,...,xn) ∈ I*n, where I* = {0,1}, let G(x1,x2,.....xn) be the subgraph of G which is obtained by deleting the vertices i1,i2,...,ij ∈ V(G) (0 ≤ j ≤ n), provided that xi1 = xi2 =..... = xij = 0. Let G[x1, x2 .... , xn] be the characteristic polynomial of G(x1, x2,.., xn), understanding that G[0,0,..., 0] ≡ 1. We prove that G[k1,k2 ....., kn] = Σx∈I*n [Πi=1n Pki+xi-2(λ)](-1)n-(Σi=1n xi) G[x1,x2 ....., xn] where X = (x1,x2,...,xn); G[k1,k2,...,kn] and Pn(λ denote the characteristic polynomial of G(k1,k2 ....,kn) and Pn, respectively. Besides, if G is a graph with λ1(G) ≥ 1 we show that λ1(G) ≤ λ1(G(k1,k2,...,kn)) λ1(G) + λ1-1(G) for all positive integers k1, k2....,kn, where λ1 denotes the largest eigenvalue.