An upper bound on the sum of squares of degrees in a hypergraph

Moments of graphs in monotone families

“…Bey [3] gives a generalization of de Caen's bound to hypergraphs. A general bound on k can be found in Székely, Clark, and Entringer [16].…”

On the Sum of Powers of the Degrees of Graphs

“…This bound was generalised to hypergraphs by Bey [1] and improved to [2] by Das [8]. De Caen's inequality was also used by Li and Pan [7] to provide an upper bound on the largest eigenvalue of the Laplacian of a graph.…”

“…It is easy to check that Theorem 1.1 (with p = 3 and q = 6) implies (1). Moreover, the implicit condition ∆(G) ≥ 2p in Theorem 1.1 is necessary.…”

The sum of the squares of degrees: Sharp asymptotics

“…an appealingly simple form." He was right-his result motivated further research, e.g., see [3,7,8]. Sadly enough, neither de Caen, nor his successors refer to the work done before Olpp. In summary, the result (4) is exact but complicated, while de Caen's result (5) is simple but inexact.…”

“…2m) 3/2 (n 2 − 2m) 3/2 + 4mn − n3 and, if 0 m n 2 /4, then(2m) 3/2 (n 2 − 2m) 3/2 + 4mn − n 3 .In other words,F (n, m) = max{(2m) 3/2 , (n 2 − 2m) 3/2 + 4mn − n 3 }.Lemma 4 implies that, for all n and m,C(n, m) m √ 8m + 1 − m (2m) 3/2 ;Lemma 5 implies that, for m n 2 /4, S(n, m) (n 2 − 2m) 3/2 + 4mn − n 3 , and so, in view of (4), the second inequality in(8)is proved.Proof of the first inequality in(8): To prove the first inequality in(8), observe that, by Proposition 3, we have (2m) 3/2 − 3m C(n, m), and so, if m n 2 /4, then F (n, m) − 4m C(n, m).…”