Characteristic polynomial and main eigenvalues of (0,1,2)–multigraphs

Let G = (V, E) be a simple graph with vertex set V and edge set E. Let H = (G, ) denote the multigraph whose underlying graph is G, where : E {1, 2, . . ., k} is a function that assigns a positive integer multiplicity to each edge.If k = 2, we refer to the multigraph H = (G, ) as a (0, 1, 2)-multigraph.An eigenvalue (of the adjacency matrix) of a (0, 1, 2)-multigraph of order n is called a main eigenvalue if there exists an associated eigenvector that is not orthogonal to the all-ones vector of dimension n.In this paper, we derive formulas for the coefficients of the characteristic polynomial of a (0, 1, 2)-multigraph.We also establish recursive formulas for computing the characteristic polynomial of a (0, 1, 2)-multigraph.Furthermore, we characterize all (0, 1, 2)-multigraphs that have exactly two main eigenvalues, under certain conditions.