Laplacian spectra of regular graph transformations

Abstract

Given a graph G with vertex set V (G) = V and edge set E(G) = E, let G l be the line graph and G c the complement of G. Let G 0 be the graph with V (G 0 ) = V and with no edges, G 1 the complete graph with the vertex set V , G + = G andand vertex v is incident (resp., not incident) to edge e in G. Given x, y, z ∈ {0, 1, +, −}, the xyz-transformation G xyz of G is the graph with the vertex set V (G xyz ) = V ∪ E and the edge set E(G xyz ) = E(G x ) ∪ E((G l ) y ) ∪ E(W ), where W = B(G) if z = +, W = B c (G) if z = −, W is the graph with V (W ) = V ∪ E and with no edges if z = 0, and W is the complete bipartite graph with parts V and E if z = 1. In this paper we obtain the Laplacian characteristic polynomials and some other Laplacian parameters of every xyz-transformation of an r-regular graph G in terms of |V |, r, and the Laplacian spectrum of G.

Computing First Zagreb index and F-index of New C-products of Graphs

“…Motivated by applications of Cartesian product of graphs, here we are more generalize the concept of Cartesian products of graphs and introduce the new C-products of graphs. For this purpose we proceed to introduce some notions and definition of [7].…”

On the Planarity of <i>G</i><sup>++−</sup>

“…Wu et al [10] proved that G −++ is planar if and only if the order of G is at most 4. We refer to [4,5,6,7,10,12,13] for more relevant results on G xyz . As usual, the complete graph, the cycle, the path of order n are denoted , , n n n K C P , respectively.…”

Spectra of digraph transformations

“…The goal of this paper is to consider (and establish some properties of) certain operations depending on parameters x, y, z ∈ {0, 1, +, −}. These operations induce functions T xyz : D → D. We put T xyz (D) = D xyz and call D xyz the xyz-transformation of D, which is similar to the xyz-transformation of an undirected graph (see, for example, [4]).…”

“…In 1967 Kelmans established the formulas for the Laplacian polynomials and the number of spanning trees of G 0++ , G 0+0 , G 00+ , and L(G) [19]. Recently, Deng, Kelmans, and Meng presented for a regular graph G and all x, y, z ∈ {0, 1, +, −} the formulas of the Laplacian polynomials and the number of spanning trees of G xyz in terms of the number of vertices, number of edges, and the Laplacian spectrum of G [4]. The zeta functions of G 0++ and G +++ and their coverings were discussed in [21].…”