Spectra of (H1,H2)-merged subdivision graph of a graph

Abstract

In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, R−graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi et al., 2017), and consequently, Q−graph, total graph, and quasitotal graph. We denote this new graph by [S(G)] H1 H2 , where G is a graph and, H 1 and H 2 are suitable graphs corresponding to G. Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of [S(G)] H1 H2 for some classes of graphs G, H 1 and H 2 . From these results, we derive the L-spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.

Spectra of generalized corona of graphs constrained by vertex subsets

“…Over the past five decades, considerable attention has been paid by the researchers on the spectra of graphs obtained using some graph operations such as union, Cartesian product, strong product, NEPS, rooted product, corona product, join, vertex deletion etc. For the results on the spectra of these graphs, we refer the reader to [5,9,13,25,26,27] and the references cited there in.…”

“…In the literature, several graph constructions have been made using one or more graphs. For the reader's convenience, here we recall the definitions of graphs constructed by some unary graph operations: The Further, the following unary graph operations are defined the spectra of the graphs obtained by them are studied in [25]: The point complete subdivision graph of G is the graph obtained by taking one copy of S(G) and joining all the vertices v i , v j ∈ V (G). The Q-complemented graph of G is the graph obtained by taking one copy of S(G) and joining the new vertices which lie on the non-adjacent edges of G. The total complemented graph of G is the graph obtained by taking one copy of R(G) and joining the new vertices lie which on the non-adjacent edges of G. The quasitotal complemented graph of G is the graph obtained by taking one copy of Q-complemented graph of G and joining all the vertices v i , v j ∈ V (G) which are not adjacent in G. The complete Q-complemented graph of G is the graph obtained by taking one copy of Q-complemented graph of G and joining all the vertices of v i , v j ∈ V (G).…”

Tensor join of hypergraphs and its spectra

“…In the literature, several graph operations were defined and the spectra of graphs constructed by these graph operations were determined; see [5,6,7,14,15,16,19,20,21] and the references therein. Recently, Gayathri and Rajkumar [8] introduced a graph operation, namely, M-join.…”