The spectrum of the edge corona of two graphs

Abstract

Abstract. Given two graphs G 1 , with vertices 1, 2, ..., n and edges e 1 , e 2 , ..., em, and G 2 , the edge corona G 1 ⋄ G 2 of G 1 and G 2 is defined as the graph obtained by taking m copies of G 2 and for each edge e k = ij of G, joining edges between the two end-vertices i, j of e k and each vertex of the k-copy of G 2 . In this paper, the adjacency spectrum and Laplacian spectrum of G 1 ⋄ G 2 are given in terms of the spectrum and Laplacian spectrum of G 1 and G 2 , respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.

The Laplacian spectrum of corona of two graphs

“…The following Theorem 4.2, first addressed in [9], is an immediate consequence of Theorem 4.1. We remark that here our method is straight-forward and different from that of Theorem 2.4.…”

“…The following two definitions come frow [2,9]. Let G 1 and G 2 be two graphs on disjoint sets of n 1 and n 2 vertices, m 1 and m 2 edges, respectively.…”

“…Barik et al [1] provided complete information about the adjacency spectrum of G 1 •G 2 for a connected graph G 1 and a regular graph G 2 , and complete information about the Laplacian spectrum of G 1 • G 2 using the spectrum (and the Laplacian spectrum, respectively) of G 1 and G 2 . In 2010, Hou and Shiu [7] considered the adjacency spectrum of G 1 G 2 for a connected regular graph G 1 and a regular graph G 2 and the Laplacian spectrum of G 1 G 2 for a connected regular graph G 1 and a graph G 2 . Recently, McLeman and McNicholas [3], by introducing a new invariant called the coronal of a graph, also discussed the spectrum of…”

Normalized Laplacian spectrum of different type of coronas of two regular graphs

“…Barik et al [2] provided complete information about the adjacency spectrum of G 1 •G 2 for a connected graph G 1 and a regular graph G 2 and the Laplacian spectrum of G 1 • G 2 for arbitrary graphs G 1 and G 2 . Hou and Shiu [10] found the adjacency spectrum of G 1 G 2 for a connected regular graph G 1 and a regular graph G 2 and the Laplacian spectrum of the same for a connected regular graph G 1 and a graph G 2 . In [14], Wang and Zhou gave complete information about the signless Laplacian spectrum of G 1 • G 2 for a graph G 1 and a regular graph G 2 and the signless Laplacian spectrum of G 1 G 2 for a connected regular graph G 1 and a regular graph G 2 .…”

“…(ii) The edge corona [10] of G 1 and G 2 , denoted by G 1 G 2 , is the graph obtained by taking one copy of G 1 and m 1 copies of G 2 , and then joining two end vertices of the i th edge of G 1 to every vertex in the i th copy of G 2 for i = 1, 2, . .…”