Graphs with two main and two plain eigenvalues

Abstract

In honor of Dragoš Cvetković on the occasion of his 75th birthday.Abstract. In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.

Some new aspects of main eigenvalues of graphs

“…Namely, in Rowlinson (2007), a complete survey on the main eigenvalues of a graph (up to 2007) is given and the graphs with exactly two main eigenvalues deserve particular attention. New results on these graphs appear in more recent publications as the works by , Shi (2009), Hu et al (2009), and Hayat et al (2016Hayat et al ( , 2017.…”

Classifying Seven-Valent Symmetric Graphs of Order 8pq

“…So p = 7 or q = 7. If p = 7, then q > 7 and q 2 |T| 2 , which contradicts with |N| 2 27 • 3 4 • 5 2 • 7 • p • q. If q = 7, then p = 5.…”

“…Otherwise, d ≥ 2, and thus 7 2 |N|. We have d = 2 as |N| 2 27 • 3 4 • 5 2 • 7 • p • q. So p = 7 or q = 7.…”

“…Note that 35 |T|. By checking the nonabelian simple group of an order less than 2 13 • 3 2 • 5 • 7, we have that T = A 7 , A 8 or PSL (3,4), and N = A 7 2 , A 8 2 or PSL(3, 4) 2 as d = 2. On the other side of the coin, C ¢ A, C ∩ N = 1 and thus ⟨C,…”

“…There are many graph parameters to characterize the reliability and vulnerability of an interconnection network, such as spectral characterization, main eigenvalues, distance characteristic polynomials, and arc-transitivity. Among these parameters, the spectral characterizations, main eigenvalues, and distance characteristic polynomials are the better ones to measure the stability of a network; see [3][4][5][6][7], for example. For arc-transitivity, see [8], as an example.…”