The split graph $K_r+\overline {K_s}$ on $r+s$ vertices is denoted by $S_{r,s}$. A non-increasing sequence $\pi =(d_1,d_2,\ldots ,d_n)$ of nonnegative integers is said to be potentially $S_{r,s}$-graphic if there exists a realization of $\pi $ containing $S_{r,s}$ as a subgraph. In this paper, we obtain a Havel-Hakimi type procedure and a simple sufficient condition for $\pi $ to be potentially $S_{r,s}$-graphic. They are extensions of two theorems due to A. R. Rao (The clique number of a graph with given degree sequence, Graph Theory, Proc. Symp., Calcutta 1976, ISI Lect. Notes Series 4 (1979), 251–267 and An Erdős-Gallai type result on the clique number of a realization of a degree sequence, unpublished).
Let $r\ge 3$, $n\ge r$ and $\pi =(d_1,d_2,\ldots ,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi $ has a realization $G$ with vertex set $V(G)=\{v_1,v_2,\ldots ,v_n\}$ such that $d_G(v_i)=d_i$ for $i=1,2,\ldots , n$ and $v_1v_2\cdots v_rv_1$ is a cycle of length $r$ in $G$, then $\pi $ is said to be potentially $C_r''$-graphic. In this paper, we give a characterization for $\pi $ to be potentially $C_r''$-graphic.
For given a graph $H$, a graphic sequence $\pi =(d_1,d_2,\ldots ,d_n)$ is said to be potentially $H$-graphic if there is a realization of $\pi $ containing $H$ as a subgraph. In this paper, we characterize the potentially $(K_5-e)$-positive graphic sequences and give two simple necessary and sufficient conditions for a positive graphic sequence $\pi $ to be potentially $K_5$-graphic, where $K_r$ is a complete graph on $r$ vertices and $K_r-e$ is a graph obtained from $K_r$ by deleting one edge. Moreover, we also give a simple necessary and sufficient condition for a positive graphic sequence $\pi $ to be potentially $K_6$-graphic.
Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_4$ and $K_5-Y_4$-graphic sequences where $Y_4$ is a tree on 5 vertices and 3 leaves.
The set of all non-increasing nonnegative integer sequences $\pi =$ ($d(v_1 ),d(v_2 ), \dots , d(v_n )$) is denoted by ${\rm NS}_n$. A sequence $\pi \in {\rm NS}_n$ 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 ${\rm NS}_n$ is denoted by ${\rm GS}_n$. 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 $\sigma (H,n)$.
In this paper, we have investigated some properties of the first Dirichlet eigenvalue of a bicyclic graph with boundary condition. These results can be used to characterize the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicyclic graphs with a given graphic bicyclic degree sequence with minor conditions. Moreover, the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicycle graphs with fixed $k$ interior vertices of degree at least 3 are obtained.