Abstract. This paper gives tight upper bounds on the largest eigenvalue q (G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle.If n is odd, let Fn be the friendship graph of order n; if n is even, let Fn be F n−1 with an extra edge hung to its center. It is shown that if G is a graph of order n ≥ 4, with no 4-cycle, thenLet S n,k be the join of a complete graph of order k and an independent set of order n − k. It is shown that if G is a graph of order n ≥ 6, with no 5-cycle, thenIt is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q (G) of graphs with forbidden cycles.

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Abstract.We characterize when a tree of diameter 4 has integer index and we provide examples of infinite families of non-integral trees with integer index. We also determine a tight upper bound for the index of any tree of diameter 4 based on its maximum degree. Moreover, we present a new infinite family of integral trees of diameter 4.

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