Some results on starlike trees and sunlike graphs

  • Mirko Lepović


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.

AMS Mathematics Subject Classification


Key words and phrases

Graph Spectrum of a graph Laplacian characteristic polynomial 


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Copyright information

© Korean Society for Computational & Applied Mathematics 2003

Authors and Affiliations

  • Mirko Lepović
    • 1
  1. 1.KragujevacYuguslavia

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