Some results on starlike trees and sunlike graphs

Abstract

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H ₁,H ₂, ...H _n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH _i with thei-th vertex ofG. In particular, ifH is the family of the paths$P_{k_1 } , P_{k_2 } , ..., P_{k_n }$ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k ₁,k ₂, …,k _n ). For any (x ₁,x ₂, …,x _n ) ∈I ⁿ_* , whereI _*={0,1}, letG(x ₁,x ₂, …,x _n ) be the subgraph ofG which is obtained by deleting the verticesi ₁, i₂, …,i _j ∈ V(G) (0≤j≤n), provided that$x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$. LetG(x ₁,x ₂,…, x _n] be the characteristic polynomial ofG(x ₁,x ₂,…, x _n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that

$$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$

where x=(x ₁,x ₂,…,x _n );G[k ₁,k ₂,…,k _n ] andP _n (γ) denote the characteristic polynomial ofG(k ₁,k ₂,…,k _n ) andP _n , respectively. Besides, ifG is a graph with λ₁(G)≥1 we show that λ₁(G)≤λ₁(G(k ₁,k ₂, ...,k _n )) < for all positive integersk ₁,k ₂,…,k _n , where λ₁ denotes the largest eigenvalue.