My (philosophical) query arises from studying the fantastic paper of Chung-Graham-Wilson the place the authors introduces the notion of quasi-random graphs.
The principle goal of the paper is to indicate that many properties of a graph are in actual fact equal. I focus on this query in two properties, particularly (I’ll use the notation of the paper):
P3 property provides info concering the second largest eigenvalue of the adjacency matrix of the graph G
the place $N_G(C_t)$ is the variety of copies of the cycle on t vertices. The paper reveals that, amongst a number of different properties, G satisfies $P_2(4)$ (See that t=Four right here) if and provided that G satisfies $P_3$.
With a purpose to show that $P_2(4)$ implies $P_3$ one can use the next simple property: the variety of 4-cycles in G is
therefore, having higher bounds from this sum would give higher bounds on the second largest component within the sum.
Within the paper the proof that $P_3$ implies $P_2(4)$ just isn’t direct, and makes use of not less than 5 intermediate implications. So my query is the next:
Is there a direct technique to sure the variety of 4-cycles immediately and simply utilizing spectral info on the graph (particularly, in regards to the second largest eigenvalue, and even its multiplicity)?
It appears to me that one thing may very well be mentioned right here, much more as a result of we’ve a direct and simple relation between the variety of 4-cycles and the spectrum of the matrix.