Spectra of the extended neighborhood corona and extended corona of two graphs

Abstract

In this paper we define extended corona and extended neighborhood corona of two graphs G 1 and G 2 , which are denoted by G 1 • G 2 and G 1 * G 2 respectively. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As applications, we give methods to construct infinite families of integral graphs, Laplacian integral graphs and expander graphs from known ones.

Constructions of new integral graph families

“…Lemma 1.1 provides a nice characterization of integral complete products of regular graphs. However, it is not easy to find families of such graphs satisfying Equation (1). Some examples are stated in the following theorem: Theorem 2.1.…”

“…There are several infinite families of integral graphs known (cf. [5,4,7,6,1]), but they appear rarely compared to the huge number of graphs at all. Thus, there is still no satisfying answer to Harary's and Schwenk's question 'Which graphs have integral spectra?'.…”

Spectra of (M,ℳ)-corona-join of graphs

“…, n [18]. Adiga and Rakshith [2] defined the following: The extended neighborhood corona (resp. extended corona) of two graphs G 1 and G 2 is the graph obtained by taking the neighborhood corona (resp.…”

Perfect codes in some products of graphs

“…This product was first introduced by Harary in 1969 [9]. We encourage the readers to consult [2,16,18,19] for more information on corona product.…”