Maxima of the Q-index, Forbidden 4-ycle and 5-cycle

Abstract

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.

Merging the A-and Q-spectral theories

“…Now, suppose that G has no isolated vertices. Select a vertex u with x u = 0 and let {u, v} ∈ E. Then we have strict inequality in (14) and so A α (G) is positive definite. ✷…”

Sum of the k Largest Signless Laplacian Eigenvalues

“…was intensively studied for several cases. In [1], de Freitas et al studied the case for graphs with no cycles of length 4 and 5, and two conjectures are formulated for the maximum q 1 (G) of even/odd cycle-free graphs.…”

Maxima of the Q-index: Forbidden even cycles

“…Before going further, we shall make three remarks. First, recall an estimate of q S + n,k given in [7], where it was shown that if k ≥ 2 and n > 5k 2 , then…”

“…Also, let S + n,k be the graph obtained by adding an edge to S n,k . In [7] the following conjecture has been raised:…”