In this paper, all connected bipartite graphs are characterized whose third largest Laplacian eigenvalue is less than three. Moreover, the result is used to characterize all connected bipartite graphs with exactly two Laplacian eigenvalues not less than three, and all connected line graphs of bipart
Some graphs with small second eigenvalue
β Scribed by Joel Friedman
- Publisher
- Springer-Verlag
- Year
- 1995
- Tongue
- English
- Weight
- 577 KB
- Volume
- 15
- Category
- Article
- ISSN
- 0209-9683
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## Abstract In this paper we prove that for a simple graph __G__ without isolated vertices 0 < Ξ»~2~(__G__) < 1/3 if and only if __G__ β KΜ~__n__β3~ V (__K__~1~ βͺ __K__~2~), the graph obtained by joining each vertex of KΜ~__n__β3~ to each vertex of __K__~1~ βͺ __K__~2~). Β© 1993 John Wiley & Sons, Inc
Nilli, A., On the second eigenvalue of a graph, Discrete Mathematics 91 (1991) 207-210. It is shown that the second largest eigenvalue of the adjacency matrix of any G containing two edges the distance between which is at least 2k + 2 is at least (2G -l)/(k + 1).
In this paper all connected line graphs whose second largest eigenvalue does not exceed 1 are characterized. Besides, all minimal line graphs with second largest eigenvalue greater than 1 are determined.
Let \(G\) be a simple graph with \(p\) vertices and let \(A(G)\) be the adjacency matrix of \(G\). The characteristic polynomial of \(G\) is the characteristic polynomial of \(A(G)\) and roots of the characteristic equation are the eigenvalues of \(G\). In this paper we compute the characteristic po