In this work,for a n-vertex p-partite graph G=G(N1,N2,…,Np)(|Ni|=ni,i=1,2,…,p),we obtain lower bound for its Kirchhoff index,and the lower bound holds if and only if G is complete p-partite graphs.

Counting formulas N[(X_1,X_2,…,X_i),k] of complete i-partite graphs;

On property M(5) of some complete multipartite graphs;

Mandatory decomposition of complete multipartite graph into cycles of lengths 3,4 and 5;

It is easy to see that Ohba s conjecture is true if and only if it is true for complete multipartite graphs.

Cylic m-cycle decompositions of the complete bipartite graphs K_(n,n) for m≡0 (mod8);

The K_(1,k～-)-factorization of complete bipartite graphs;

Vertex-distinguishing VE-total colorings of wheels,fans and complete bipartite graphs K_(1,n) and K_(2,n)

Crossing number of the complete tripartite graph K_(1,10,n);

In this papaer we prove that if Zarankiewicz s conjecture is true for the case m = 7, then the crossing number of the complete tripartite graph K1,6,n is 9 [n/2][n-1/2] + 6[n/2].

Chromatic uniqueness of a kind of complete tripartite graphs;

Let K(m,n,r) denote a complete tripartite graph,this paper proved that the complete tripartite graph K(m,m,m+k) is chromatically unique if m≥2 and k≥0,K(m,m+1,m+k) is chromatically unique if m≥2 and m+1>k≥0.

In this paper the definitions of lower integral sum graph and the lower integral sum number of a graph are introduced,some properties of lower integral sum graphs are given,and it is proved that the lower integral sum number of complete tripartite graph K_ m,n,q (m,n,q≥2) is 2.