Shengbiao Hu

Bio: Shengbiao Hu is an academic researcher from Qinghai University for Nationalities. The author has contributed to research in topics: Vertex (geometry) & Bound graph. The author has an hindex of 1, co-authored 1 publications receiving 3 citations.

A sharp lower bound of the spectral radius of simple graphs

Abstract: Let G be a simple connected graph with n vertices and let p(G) be its spectral radius. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i. Let Ni = Σj~i tj and Mi = Σj~i Nj. We find a sharp lower bound of p(G), which only contains two parameter Ni and Mi. Our result extends recent known results.

Sharp Upper Bounds for Energy and Randic Energy

Abstract: Sharp upper bounds for the energy and Randic energy of a (bipartite) graph are established. From these, some previously known results could be deduced.

Inequalities for the number of walks in graphs

Abstract: We investigate the growth of the number wk of walks of length k in undirected graphs as well as related inequalities. In the first part, we derive the inequalities w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) and w2a+c(v, v) · w2(a+b)+c(v, v) ≤ w2a(v, v) · w2(a+b+c)(v, v) for the number wk(v, v) of closed walks of length k starting at a given vertex v. The first is a direct implication of a matrix inequality by Marcus and Newman and generalizes two inequalities by Lagarias et al. and Dress & Gutman. We then use an inequality of Blakley and Dixon to show the inequality wk2e+p ≤ w2e+pk · wk−12e which also generalizes the inequality by Dress and Gutman and also an inequality by Erdos and Simonovits. Both results can be translated directly into the corresponding forms using the higher order densities, which extends former results. In the second part, we provide a new family of lower bounds for the largest eigenvalue λ1 of the adjacency matrix based on closed walks and apply the before mentioned inequalities to show monotonicity in this and a related family of lower bounds of Nikiforov. This leads to generalized upper bounds for the energy of graphs. In the third part, we demonstrate that a further natural generalization of the inequality w2a+c · w2(a+b)+c ≤ w2a · w2(a+b+c) is not valid for general graphs. We show that wa+b · wa+b+c ≤ wa · wa+2b+c does not hold even in very restricted cases like w1 · w2 ≤ w0 · w3 (i.e., d · w2 ≤ w3) in the context of bipartite or cycle free graphs. In contrast, we show that surprisingly this inequality is always satisfied for trees and show how to construct worst-case instances (regarding the difference of both sides of the inequality) for a given degree sequence. We also provide a proof for the inequality w1 · w4 ≤ w0 · w5 (i.e., d · w4 ≤ w5) for trees and conclude with a corresponding conjecture for longer walks.

Further Results on the Number of Walks in Graphs and Weighted Entry Sums of Matrix Powers

Abstract: We consider the number of walks in undirected and directed graphs and, more generally, the weighted sum of entries of matrix powers. In this respect, we generalize an earlier result for Hermitian matrices. By using these inequalities for the entry sum of matrix powers, we deduce similar inequalities for iterated kernels. For further conceivable inequalities, we provide counterexamples in the form of graphs that contradict the corresponding statement for the number of walks. For the largest eigenvalue of adjacency matrices, we generalize a bound of Nikiforov that uses the number of walks. Furthermore, we relate the number of walks in graphs to the number of nodes and the number of edges in iterated directed line graphs.