Elsevier

Discrete Mathematics

Volume 307, Issue 2, 28 January 2007, Pages 280-284
Discrete Mathematics

Note
The largest eigenvalue of unicyclic graphs

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Abstract

Let G be a simple graph. Let λ1(G) and μ1(G) denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let Δ denote the largest vertex degree. If G has just one cycle, thenλ1(G)2Δ-1.The equality holds if and only if GCn.

Andμ1(G)Δ+2Δ-1.The equality holds if and only if GCn, n is even.

MSC

05C50
05C05

Keywords

Unicyclic graph
Adjacency matrix
Laplacian matrix
Largest eigenvalue

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Supported by the Key Project of Chinese Ministry of Education No. 205169.