Abstract
Let G be a connected non-regular non-bipartite graph whose adjacency matrix has spectrum p,?(k),?(l), where k,l ? N and p > ? > ?. We show that if ? is non-main then ?(G) ? 1 + ? - ??, with equality if and only if G is of one of three types, derived from a strongly regular graph, a symmetric design or a quasi-symmetric design (with appropriate parameters in each case).
Highlights
Let G be a graph of order n with (0, 1)-adjacency matrix A
We say that G is an integral graph if every eigenvalue of G is an integer; and G is a biregular graph if it has just two different degrees
Let C1 be the class of connected graphs with just three distinct eigenvalues, and let C2 be the class of connected graphs with exactly two main eigenvalues
Summary
In honour of Dragos Cvetkovic on the occasion of his 75th birthday. Let G be a connected non-regular non-bipartite graph whose adjacency matrix has spectrum ρ, μ(k), λ(l), where k, l ∈ IN and ρ > μ > λ. We show that if μ is non-main δ(G) ≥ 1 + μ − λμ, with equality if and only if G is of one of three types, derived from a strongly regular graph, a symmetric design or a quasi-symmetric design (with appropriate parameters in each case)
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