# Some results on starlike trees and sunlike graphs

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## 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 trees where x=

*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 letH={*H*_{1},*H*_{2}, ...*H*_{ n }} be a family of rooted graphs. According to [2], the rooted product*G*(H) is the graph obtained by identifying the root of*H*_{ i }with the*i*-th vertex of*G*. 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 graph*G*(H) is called the sunlike graph and is denoted by*G(k*_{1},*k*_{2}, …,*k*_{ n }). For any (*x*_{1},*x*_{2}, …,*x*_{ n }) ∈*I*_{*}^{ n }, where*I*_{*}={0,1}, let*G(x*_{1},*x*_{2}, …,*x*_{ n }) be the subgraph of*G*which is obtained by deleting the vertices*i*_{1}, i_{2}, …,*i*_{ j }*∈ V(G) (0≤j≤n)*, provided that\(x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0\). Let*G(x*_{1},*x*_{2},*…, x*_{n}] be the characteristic polynomial of*G(x*_{1},*x*_{2},*…, x*_{ n }), understanding that*G*[0, 0, …, 0] ≡ 1. We prove that$$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$

*(x*_{1},*x*_{2},*…,x*_{ n });*G[k*_{1},*k*_{2},*…,k*_{ n }] and*P*_{ n }*(γ)*denote the characteristic polynomial of*G(k*_{1},*k*_{2},*…,k*_{ n }) and*P*_{ n }, respectively. Besides, if*G*is a graph with λ_{1}(*G*)≥1 we show that λ_{1}(*G*)≤λ_{1}(*G*(*k*_{1},*k*_{2}, ...,*k*_{ n })) < for all positive integers*k*_{1},*k*_{2},*…,k*_{ n }, where λ_{1}denotes the largest eigenvalue.## AMS Mathematics Subject Classification

05C50## Key words and phrases

Graph Spectrum of a graph Laplacian characteristic polynomial## References

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© Korean Society for Computational & Applied Mathematics 2003