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.
On the second eigenvalue of a graph
β Scribed by A. Nilli
- Publisher
- Elsevier Science
- Year
- 1991
- Tongue
- English
- Weight
- 241 KB
- Volume
- 91
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
β¦ Synopsis
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).
π SIMILAR VOLUMES
## 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
## Abstract It is well known that the smallest eigenvalue of the adjacency matrix of a connected __d__βregular graph is at least β __d__ and is strictly greater than β __d__ if the graph is not bipartite. More generally, for any connected graph __G = (V, E)__, consider the matrix __Q = D + A__ wher
A graph is called of type k if it is connected, regular, and has k distinct eigenvalues. For example graphs of type 2 are the complete graphs, while those of type 3 are the strongly regular graphs. We prove that for any positive integer n, every graph can be embedded in n cospectral, non-isomorphic