I hope to know some good references about the use of integrals to study the graph theory:
For example, it seems that $$ \int^{\infty}_{-\infty} dx \exp(-x^2/2+\lambda x^3/3!) $$ whose coefficients in the $V$ powers of $\lambda$, the coefficient of $\lambda^V$ counts the possible trivalent graphs of $V$ vertices.
For example, it seems that $$ \int dM \exp(\text{Tr}(-M^2/2+\lambda M^3/3!)) $$ where $M$ is a rank-$N$ Hermitian matrix. The coefficient of $N^{2-2g}\lambda^V$ counts the possible trivalent graphs of $V$ vertices on a genus-$g$ Riemann surfaces.
Are there some similar or more general statements of such integrals in the graph theory? Any References are welcome.
Some intuitive way to obtain the above formulas?
In paper I was reading recently, the author said "it is well known that many enumerative problems in graph theory can be solved using Gaussian integrals," and cited the following book:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004, With an appendix by Don B. Zagier, Low-Dimensional Topology, II. MR 2036721. (Springer, or Springer Link).
Chapter 3 discusses matrix integral methods. It gives some examples, for instance a matrix integral for enumerating $1$-vertex graphs that fill a surface of genus $g$, or enumerating planar $4$-regular graphs that are the generic image of a circle. Perhaps the book describes the techniques well enough that you can figure out the construction of your given integrals.
A note to Proposition 3.2.10 mentions how you can disregard genus information by setting the matrix size to $1\times 1$. This appears to be the transformation from your second integral to your first. Given what I've picked up so far, perhaps it ought to be "where $M$ is an $n\times n$ Hermitian matrix" rather than "where $M$ is a rank-$n$ Hermitian matrix."
After looking at a paper by one of the book's authors,
A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Mathematical and Computer Modelling 26 (1997), no. 8-10, 281–304, Combinatorics and physics (Marseilles, 1995). MR 1492512
I think I can tell you what the terms of the integral represent. The first part is that a Gaussian measure on the space of $n\times n$ Hermitian matrices is given by $d\mu(M)=\exp(\operatorname{tr}(-M^2/2))dM$, though in the paper equation (6) has some more normalization terms. The second part is $$\exp(\operatorname{tr}(\lambda M^3/3!))=\sum_{k=0}^{\infty}\operatorname{tr}(\lambda M^3/3!)^k/k!=\sum_{k=0}^\infty(\lambda/3!)^{nk}\operatorname{tr}(M^3)^k/k!.$$ I do not understand what the $3!$ is doing in there, except perhaps to remove the dihedral symmetry of a $3$-valent vertex. The $k!$ is likely to remove the vertex-renaming symmetry. The paper says that $\int \operatorname{tr}(M^3)^k\,d\mu$ expanded in terms of $N$ has coefficients giving the number of $3$-valent graphs with $k$ vertices in a particular genus. Integrating the infinite sum makes it so the coefficients of $N$ are power series in $\lambda$, the coefficients of which give the number of graphs of a particular number of vertices.