A characterization of the smallest eigenvalue of a graph

It is well known that the smallest eigenvalue of the adjacency matrix of a connected ¿-regular graph is at least -d and is strictly greater than -d if the graph is not bipartite. More generally, for any connected graph G = (V JE), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G , and A is the adjacency matrix of G . Then Q is positive semi-definite, and the smallest eigenvalue of Q is 0 if and only if G is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of non-bipartiteness of G. For any we denote by emin(S) the minimum number of edges that need to be removed from the induced subgraph on S to make it bipartite. Also, we denote by cut(S) the set of edges with one end in S and the other in V S . We define the parameter \\r as e min(S)+ \cut (5)| The parameter y is a measure of the non-bipartiteness of the graph G . We will show that the smallest eigen­ value of Q is bounded above and below by functions of \y. For ¿-regular graphs, this characterizes the separa­ tion of the smallest eigenvalue of the adjacency matrix from -¿ . These results can be easily extended to weighted graphs.