Unlike particles in the classical dynamics, the dynamical behavior of a complex network maybe not synchronized but fragmented, even a heterogenous moving in the eyes of human beings, which finally results in characterizing a complex network by random method or probability with statistics sometimes. However, such a dynamics on complex network is quite different from dynamics on particles because all mathematics are established on compatible systems but none on a heterogenous one. Naturally, a heterogenous system produces a contradictory system in general which was abandoned in classical mathematics but exists everywhere, i.e., it is inevitable if we would like to understand the reality of things in the world. Thus, we should establish such a mathematics on those of elements that contradictions appear together peacefully but without loss of the individual characters. For this objective, the network or in general, the continuity flow is the best candidate of the element, i.e., mathematical elements over a topological graph G in space. The main purpose of this paper is to establish such a mathematical theory on networks, including algebraic operations, differential and integral operations on networks, G-isomorphic operators, i.e., network mappings remains the unchanged underlying graph G with a generalization of the fundamental theorem of calculus, algebraic or differential equations with flow solutions and also, the dynamical equations of network with applications to other sciences by e-indexes on network. All of these results show the importance, i.e., quantitatively characterizing the reality of things by mathematical combinatorics.

Actually, one establishes mathematicalmodel for understanding a natural thing or matter $T$ by its mathematical property $\widehat{T}$ characterized by model, called mathematical reality. {\it Could we always conclude the equality $\widehat{T}=T$ in nature}? The answer is disappointing by Godel's incomplete theorem which claims that any formal mathematical axiom system is incomplete because it always has one proposition that can neither be proved, nor disproved in this system. Thus, we can not determine $\widehat{T}=T$ or $\not=T$ sometimes by the boundary of mathematics. Generally, a natural thing or matter is complex, even hybrid with other things. Unlike purely thinking, physics and life science determine natural things by subdividing them into irreducible but detectable units such as those of quarks, gluons or cells, i.e., the composition theory of $T$ in the microcosmic level, which concludes the reality of $T$ is the whole behavior of a complex network induced by local units. However, all mathematical elements can only determines the character of $T$ locally and usually brings about a contradictory system in mathematics. {\it Could we establish a mathematics on complex networks avoiding Godel's incomplete theorem for science, i.e., mathematical combinatorics}? The answer is positive motivated by the traditional Chinese medicine, in which a living person is completely reflected by $12$ meridians with balance of Yin ($Y^-$) and Yang ($Y^+$) on his body, which alludes that there is a new kind of mathematical elements, called {\it harmonic flows} $\overrightarrow{G}^{L^2}$ with edge labeled by $L^2:(v,u)\in E\left(\overrightarrow{G}\right)\rightarrow L(v,u)-iL(v,u))$, where $i^2=-1$, $L(v,u)\in\mathscr{B}$ and $2$ end-operators $A_{vu}^+, A_{vu}^-$ on Banach space $\mathscr{B}$ holding with the continuity equation on vertices $v\in V\left(\overrightarrow{G}\right)$ with dynamic behavior characterized by Euler-Lagrange equations.

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.

There are no theory on antimatter structure unless the mirror of its normal matter, with the same mass but opposite qualities such as electric charge, spin, , etc. to its matter counterparts holding with the Standard Model of Particle. In theory, a matter will be immediately annihilated if it meets with its antimatter, leaving nothing unless energy behind, and the amounts of matter with that of antimatter should be created equally in the Big Bang. So, none of us should exist in principle but we are indeed existing. A few physicists explain this puzzling thing by technical assuming there were extra matter particles for every billion matter-antimatter pairs, or asymmetry of matter and antimatter in the end. Certainly, this assumption comes into beings by a priori hypothesis that the matter and antimatter forming both complying with a same composition mechanism after the Big Bang, i.e., antimatter consists of antimolecules, antimolecule consists of antiatoms and antiatom consists of antielectrons, antiprotons and antineutrons without experimental evidences unless the antihydrogen, only one antimolecule. Why only these antimatters are detected by experiments? Are there all antimatters in the universe? In fact, if the behavior of gluon in antimatter, i.e., antigluon is not like the behavior but opposites to its matter counterparts or reverses gluon interaction Fgk to