Some New Properties for Degree-Based Graph Entropies

Abstract

Abstract:The graph entropies inspired by Shannon's entropy concept become the information-theoretic quantities for measuring the structural information of graphs and complex networks. In this paper, we continue studying some new properties of the graph entropies based on information functionals involving vertex degrees. We prove the monotonicity of the graph entropies with respect to the power exponent. Considering only the maximum and minimum degrees of the (n, m)-graph, we obtain some upper and lower bounds for the degree-based graph entropy. These bounds have different performances to restrict the degree-based graph entropy in different kinds of graphs. Moreover the degree-based graph entropy can be estimated by these bounds.

New Upper Bound and Lower Bound for Degree-Based Network Entropy

“…Relatively speaking, the entropy is nearer to the lower bound L NE than to the upper bound U NE . Compared with the list for them in Table 3 in [33], we also find the new lower bound is better. …”

“…The new method derives new upper bound and lower bound. In the results of analyzing the graph examples and the network extracted from the computer network, we find the new lower bound is better than the earlier ones in [33] to monocentric homogeneous dendrimer graph and the new upper bound performs better in the extracted computer network. This paper is organized as follows: In Section 2, some notations in graph theory and the degree-based network entropy we are going to study are introduced.…”

“…In [33] we present the earlier work that we have done. We prove the monotonicity of the entropy with respect to the power index.…”

Generalized Degree-Based Graph Entropies

“…S n is denoted a star graph characterized as a tree in which the degree of all but one vertex is 1. More details can be seen in [17,18].…”

“…To study more properties of graph entropies based on the degree powers, Lu et al obtained some upper and lower bounds which have different performances to bound the graph entropies in different kinds of graphs and showed their applications in structural complexity analysis [17,18]. Inspired by Dehmer and Mowshowitz [13], we focus on the relationships between degree powers and the parametric complexity measures and then we construct generalized degree-based graph entropies by using the concept of the mentioned generalized graph entropies.…”