Physica A 447 (2016) 355–378 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa The H 0 function, a new index for detecting structural/topological complexity information in undirected graphs Massimo Buscema a,b,∗ , Masoud Asadi-Zeydabadi b,c , Weldon Lodwick b , Marco Breda a a Semeion Research Centre of Sciences of Communication, Via Sersale 117, Rome, 00128, Italy b Department of Mathematical and Statistical Science, CCMB, University of Colorado Denver, P.O. Box 173364, Denver, CO 80217, USA c Department of Physics, University of Colorado Denver, P.O. Box 173364, Denver, CO 80217, USA highlights • New index to measure the complexity of connected undirected graphs. • Lowest zero value with an infinite chain of nodes. • Highest two value with an infinite number of nodes full connected. • New algorithm to determine graph similarity, also with different number of nodes. • Different levels of similarity: normal, weak and strong. article info Article history: Received 28 April 2015 Received in revised form 12 October 2015 Available online 23 December 2015 Keywords: Graph complexity Graph index Graph pruning abstract Significant applications such as the analysis of Alzheimer’s disease differentiated from de- mentia, or in data mining of social media, or in extracting information of drug cartel struc- tural composition, are often modeled as graphs. The structural or topological complexity or lack of it in a graph is quite often useful in understanding and more importantly, re- solving the problem. We are proposing a new index we call the H 0 function to measure the structural/topological complexity of a graph. To do this, we introduce the concept of graph pruning and its associated algorithm that is used in the development of our measure. We illustrate the behavior of our measure, the H 0 function, through different examples found in the appendix. These examples indicate that the H 0 function contains information that is useful and important characteristics of a graph. Here, we restrict ourselves to undirected. © 2015 Elsevier B.V. All rights reserved. 1. Introduction The H 0 function is a new index to measure topological information. By topological information of any undirected graph we mean information such as structural complexity, size measures, norms, that are possible to extract from a graph without any references to its geometry [1,2]. The H 0 function is a global index describing the topological complexity of an entire graph. It is especially useful for the analysis of networks arising from physical or biological systems. In much of the literature many local statistical indices are presented and used to make explicit all the hidden information that is embedded in networks [3–6], ∗ Corresponding author at: Semeion Research Centre of Sciences of Communication, Via Sersale 117, Rome, 00128, Italy. E-mail addresses: m.buscema@semeion.it, semeion@semeion.it (M. Buscema). http://dx.doi.org/10.1016/j.physa.2015.12.055 0378-4371/© 2015 Elsevier B.V. All rights reserved.