The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,1) and CD(K,n).

The set of all non-increasing nonnegative integers sequence $\pi=$ ($d(v_1 ),$ $d(v_2 ),$ $...,$ $d(v_n )$) is denoted by $NS_n$. A sequence $\pi\in NS_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $\pi$. The set of all graphic sequences in $NS_n$ is denoted by $GS_n$. A graphical sequence $\pi$ is potentially $H$-graphical if there is a realization of $\pi$ containing $H$ as a subgraph, while $\pi$ is forcibly $H$-graphical if every realization of $\pi$ contains $H$ as a subgraph. Let $K_k$ denote a complete graph on $k$ vertices. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). This paper summarizes briefly some recent results on potentially $K_{m}-G$-graphic sequences and give a useful classification for determining $\sigma(H,n)$.

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We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.

Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \bigcup P_3$ but not contains a cycle on $4$ vertices and not contains $Z_4$.