International J.Mathematical Combinatorics，2020，Vol.4（2019）：1-18
In the view of modern science, a matter is nothing else but a complex network G, i.e., the reality of matter is characterized by complex network. However, there are no such a mathematical theory on complex network unless local and statistical results. Could we establish such a mathematics on complex network? The answer is affirmative, i.e., mathematical combinatorics or mathematics over topological graphs. Then, what is a graph? How does it appears in the universe? And what is its role for understanding of the reality of matters? The main purpose of this paper is to survey the progressing process and explains the notion from graphs to complex network and then, abstracts mathematical elements for understanding reality of matters. For example, L.Euler’s solving on the problem of Kongsberg seven bridges resulted in graph theory and embedding graphs in compact n-manifold, particularly, compact 2-manifold or surface with combinatorial maps and then, complex networks with reality of matters. We introduce 2 kinds of mathematical elements respectively on living body or non-living body for self-adaptive systems in the universe, i.e., continuity flow and harmonic flow G which are essentially elements in Banach space over graphs with operator actions on ends of edges in graph G. We explain how to establish mathematics on the 2 kinds of elements, i.e., vectors underling a combinatorial structure G by generalize a few well-known theorems on Banach or Hilbert space and contribute mathematics on complex networks. All of these imply that graphs expand the mathematical field, establish the foundation on holding on the nature and networks are closer more to the real but without a systematic theory. However, its generalization enables one to establish mathematics over graphs, i.e., mathematical combinatorics on reality of matters in the universe.