Two Shorter Proofs in Spectral Graph Theory
نویسندگان
چکیده
Let G be a simple graph with V (E) as its vertex (respectively, edge) set. The spectrum of G is the spectrum of its adjacency matrix. For all other definitions (or notation not given here), especially those related to graph spectra, see [1]. The purpose of this note is to provide shorter proofs of two inequalities already known in spectral graph theory. The first bounds vertex eccentricities of a connected graph in terms of some spectral quantities, while the second bounds the spectral radius of a graph in terms of vertex degrees. The first inequality is a slight generalization of a bound due to C.D. Godsil, while the second, also known as the Runge-Hofmeister conjecture, was first proved by A.J. Hoffman et al. (The proofs given here were obtained by the first and second author, respectively.)
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