Sharp lower bounds of the least eigenvalue of planar graphs

AbstractLet G be a simple graph with n⩾3 vertices and orientable genus g and non-orientable genus h. We define the Euler characteristic χ(G) of a graph G by χ(G)=max{2−2g,2−h}. Let λ(G) be the least eigenvalue of the adjacency matrix A of G. In this paper, we obtain the following lower bounds of λ(G)λ(G)⩾−2(n−χ(G)).In particular, if G is the planar graph, thenλ(G)⩾−2n−4the equality holds if and only if G≅K2,n−2. Further, we have same result of series–parallel graph