Proof of a conjecture on the complete split-like graphs

The eigenvalues of a graph G are the eigenvalues of its adjacency matrix A(G). An eigenvalue of a graph G is said to be a main eigenvalue if it has an eigenvector not orthogonal to the main vector j = (1, 1, . . ., 1)'. A connected n-vertex biregular graph is said to be a complete split-like graph, denoted by KSL(n, q,δ), if it has q ≥ 1 universal vertices with degree n - 1 and n - q vertices with degree δ. Réti [On some properties of graph irregularity indices with a particular regard to the <r-index, Applied Math. Comput. 344-345 (2019) 107-115] gave a conjecture as follows: Let KSL(n, q, δ) be a complete split-like graph with a spectral radius A. Then KSL(n, q, δ) has exactly two main eigenvalues λ and μ., where μ < λ. In this paper we confirm the above conjecture. Moreover, we mention more general result.