On potentially $K_{r+1}-U$-graphical Sequences

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 $ngeq 5r+18, r+1 geq k geq 7,$ $j geq 6$ 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 $Z_4$. There are a number of graphs 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 $Z_4$. (for example, $C_3bigcup C_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 5)$, $C_3bigcup P_{i_1} bigcup P_{i_2} bigcup ... bigcup P_{i_p}$ $(i_1 geq 3)$, $C_3bigcup P_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 3)$, etc)

A Simple Probabilistic Algorithm for Detecting Community Structure in Social Networks

In this paper, we propose a novel semi-parametric probabilistic model which considers interactions between different communities and can provide more information about the network topology besides correctly detecting communities. By using an additional parameter, our model can not only detect community structure but also detect pattern which is a generalization of common sense network community structure. The prior parameter in our model reveals the characteristics of patterns inside the network. Results on some widely known data sets prove the efficiency of our model.