The reality of a thing is its state of existed, exists, or will exist in the world, independent on the understanding of human beings, which implies that the reality holds on by human beings is local or gradual, and mainly the Mathematical reality, not the reality of a thing. Is our mathematical theory can already be used for understanding the reality of all things in the world? The answer is not because one can not holds on the reality in many fields. For examples, the elementary particle system or ecological system, in which there are no a classical mathematical subfield applicable, i.e., a huge challenge now is appearing in front of modern mathematicians: To establish new mathematics adapting the holds on the reality of things. I research mathematics with reality beginning from 2003 and then published papers on fields, such as those of complex system and network, interaction system, contradictory system, biological populations, non-solvable differential equations, and elementary established an entirely new envelope theory for this objective by flows, i.e., mathematical combinatorics, or mathematics over graphs, which is an appropriated way for understanding the reality of a thing because it is complex, even contradictory. This book collects my mainly papers on mathematics with reality of a thing from 2007 – 2017 and most of them are the plenary or invited reports in international conferences.

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.