Applications of Ramsey Style Theorems to Eigenvalues of Graphs

Abstract

Let G be a graph, A(G) its adjacency matrix, i.e. A = (a_ij) is given by

$$ {a_{{ij}}} = \left\{ {\begin{array}{*{20}{c}} {1\,if\,i\,and\,j\,are\,adjacent\,vertices} \\ {0\,otherwise} \\ \end{array} } \right. $$

Thus, A = A(G) is a symmetric matrix whose entries are 0 and 1, with every a_ii= 0. For any real symmetric A, we denote its eigenvalues by

$$ {\lambda_1}(A) \geqslant {\lambda_2}(A) \geqslant \ldots $$

or

$$ {\lambda^1}(A) \leqslant {\lambda^2}(A) \leqslant \ldots $$

as is convenient. For A = A(G), we sometimes write λ_i(G) or λⁱ (G) for λ_i(A(G)) or λⁱ (A(G)) respectively.

The preparation of this manuscript was supported (in part) by US Army contract # DAHC04-72-C-0023.