Graphs with Eigenvalues at Least -2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP Vijaya Kumar zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED School of Mathematics TZFR Colaba, Bombay - 400005, India S. B. Rao Department of Mathematics The Ohio State University Columbus, Ohio 43210 and Statistics, Math Division Zndian Statistical Institute Calcutta-70035, India and N. M. Singhi School of Mathematics TZFR Coluba, Bomba y-400005, India Submitted by W. Bridges A BSTRA CT The family of minimal forbidden graphs for the set of graphs with all eigenvalues at least - 2 is described. It is shown that each minimal forbidden graph has at most 10 vertices and the bound is the best possible. 1. INTRODUCTION A graph is a pair G = (X, ( , )) where X is a finite set (called the set of vertices)and( , ):XxX-{O,l} satisfying(x,y)=(y,x), (x,x)=Ofor alI x, y in X. If for x, y in X, (x, y) = 1, we say that the set {x, y} is an edge of the graph G; and if ITS APPLICATIONS 46~27-42 (1982) 27 ‘?J Elsevier Science Publishing Co., Inc., 1982 52 Vanderbilt Ave., New York, NY 10017 0024.3795/82,‘050027 + 16$02.75