Coronae Graphs and Their $$\alpha $$-Eigenvalues

The eccentricity matrix 𝔼(G) of a graph G is constructed from the distance matrix D(G) by only retaining the leading values in each row and each column and setting the rest of the elements in the corresponding row and column to zero. The article aims to study the concepts of energy and the inertia of the eccentricity matrices of graphs obtained by means of various graph products, like the strong product, the Cartesian product, and the Kronecker product, and well-established relations are derived in terms of their base graphs. Furthermore, we construct infinitely many pairs of non-isomorphic graphs having the same set of eigenvalues with respect to the eccentricity matrix.

This paper considers a certain fairly large class of graph invariants and shows that for any invariant in this class, there is a pair of nonisomorphic graphs that have the same invariant. Some explicit examples of such invariants are discussed.

Let GS be the graph of order n and containing σ self-loops. The energy E(GS) of graph GS is defined as E(GS)= Σni=1|λi- σ/n|,where λ1, λ2,…, λn, be the eigenvalues of the adjacency matrix of GS. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if they have same energy. The proposed research is an effort to expand the concept of equienergetic graphs from simple graphs to graphs having self-loops. In the present work, we have obtained a pair of equienergetic graphs and the energy of complete graphs as well as complete bipartite graphs with self loops.