Forbidden theta graph, bounded spectral radius and size of non-bipartite graphs

. Zhai and Lin recently proved that if G is an n-vertex connected 8(1, 2, r + 1)-free graph, then for odd r and n 10r, or for even r and n 7r, one has rho(G) <= L n24 RIGHT FLOOR, and equality holds if and only if G is K left ceiling n2 right ceiling ,L n2 RIGHT FLOOR. In this paper, for large enough n, we prove a sharp upper bound for the spectral radius in an n-vertex H-free non-bipartite graph, where H is 8(1,2, 3) or 8(1,2, 4), and we characterize all the extremal graphs. Furthermore, for n 137, we determine the maximum number of edges in an n-vertex 8(1,2, 4)-free non-bipartite graph and characterize the unique extremal graph.