A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine $σ(K_{r+1}-(kP_2\bigcup tK_2), n)$ for $n\geq 4r+10, r+1 \geq 3k+2t, k+t \geq 2,k \geq 1, t \geq 0$ .
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 $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $σ(K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices a…
A sequence $S$ is potentially $K_{m}-C_{4}$-graphical if it has a realization containing a $K_{m}-C_{4}$ as a subgraph. Let $σ(K_{m}-C_{4}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-C_{4}, n)$ is potentially $K_{m}-C_{4}$-graphical. In this paper, we prove that $σ(K_{m}-C_{4}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq 4.$ We conjecture that equality holds for $n \geq m \geq 4.$ We prove that this conjecture is true for $m=5$.
A sequence $S$ is potentially $K_{m}-P_{k}$ graphical if it has a realization containing a $K_{m}-P_{k}$ as a subgraph. Let $σ(K_{m}-P_{k}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-P_{k}, n)$ is potentially $K_{m}-P_{k}$ graphical. In this paper, we prove that $σ(K_{m}-P_{k}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq k+1\geq 4.$ We conjecture that equality holds for $n \geq m \geq k+1\geq 4.$ We prove that this conjecture is true for $m=k+1=5$ and $m=k+2=5$.
A sequence $S$ is potentially $K_{p,1,1}$ graphical if it has a realization containing a $K_{p,1,1}$ as a subgraph, where $K_{p,1,1}$ is a complete 3-partite graph with partition sizes $p,1,1$. Let $σ(K_{p,1,1}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{p,1,1}, n)$ is potentially $K_{p,1,1}$ graphical. In this paper, we prove that $σ(K_{p,1,1}, n)\geq 2[((p+1)(n-1)+2)/2]$ for $n \geq p+2.$ We conjecture that equality holds for $n \geq 2p+4.$ We prove that this conjecture is true for $p=3$.
A graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the Laplacian operator have mean value zero. In this paper, we show that extremal independent sets make extremal graphical designs, that is, a design on which the maximum possible number of eigenfunctions have mean value zero. We then provide examples of such graphs and sets, which arise naturally in extremal combinatorics. We also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs, and provide examples for them as well. We investigate the behavior of gra…
A sequence $S$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical if it has a realization containing a $K_{p_{1},p_{2},...,p_{t}}$ as a subgraph, where $K_{p_{1},p_{2},...,p_{t}}$ is a complete t-partite graph with partition sizes $p_{1},p_{2},...,p_{t} (p_{1}\geq p_{2}\geq ...\geq p_{t} \geq 1)$. Let $σ(K_{p_{1},p_{2},...,p_{t}}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{p_{1},p_{2},...,p_{t}}, n)$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical. In this paper, we prove that $σ(K_{p_{1},p_{2},...,p_{t}}, n)\geq 2[((2p_{1}+2p_…
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.
Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. 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 $K_{m}$). A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we d…
The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular graphic bicyclic sequences. Moreover, we obtain the relations of the second Zagreb indices among the extremal graphs with different degree sequences.
A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$ is an 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $π=(d_1,\ldots,d_n)$ of nonnegative integers is a {\it graphic sequence} if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795--802) proved that if $k\ge 2$, $n\ge \frac{9}{2}k^2+\frac{19}{2}k$ and $π=(d_1,\ldots,d_n)$ is a graphic sequence with $\sum\limits_{i=1}^n d_i>(k-2)n$, then $π$ has a re…
A graphic sequence $π$ is potentially $H$-graphic if there is some realization of $π$ that contains $H$ as a subgraph. The Erdős-Jacobson-Lehel problem asks to determine $σ(H,n)$, the minimum even integer such that any $n$-term graphic sequence $π$ with sum at least $σ(H,n)$ is potentially $H$-graphic. The parameter $σ(H,n)$ is known as the potential function of $H$, and can be viewed as a degree sequence variant of the classical extremal function ${\rm ex}(n,H)$. Recently, Ferrara, LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree sequence is potentially $H$-graphic…
In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $σ(S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$-term graphical sequence $S$ having $σ(S)\ge l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $σ(H, n)$. We show $σ(C_{2m+1}, n)=m(2n-m-1)+2$, for $m\ge 3$, $n\ge 3m$; $σ(C_{2m+2}, n)=m(2n-m-1)+4$, for $m\ge 3, n\ge 5m-2$.
Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, ..., A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$. Clique partitions are related to design theory, coding theory, projective geometry, and extremal combinatorics. Let $\cp(n,r)$ denote the minimum size of a clique partition of ${[n] \choose r}$. A classical theorem of de Bruijn and Erd\H os states that $\cp(n, 2) = n$ and also determines the extremal configurations. In this paper we study $\cp(n,r)$, and show in general that for each fi…
A sequence of nonnegative integers $π$ is {\it graphic} if it is the degree sequence of some graph $G$. In this case we say that $G$ is a \textit{realization} of $π$, and we write $π=π(G)$. A graphic sequence $π$ is {\it potentially $H$-graphic} if there is a realization of $π$ that contains $H$ as a subgraph. Given nonincreasing graphic sequences $π_1=(d_1,\ldots,d_n)$ and $π_2 = (s_1,\ldots,s_n)$, we say that $π_1$ {\it majorizes} $π_2$ if $d_i \geq s_i$ for all $i$, $1 \leq i \leq n$. In 1970, Erdős showed that for any $K_{r+1}$-free graph $H$, there exists an $r$-partite graph $G$ such tha…
This survey is two-fold. We first report new progress on the spectral extremal results on the Turán type problems in graph theory. More precisely, we shall summarize the spectral Turán function in terms of the adjacency spectral radius and the signless Laplacian spectral radius for various graphs. For instance, the complete graphs, general graphs with chromatic number at least three, complete bipartite graphs, odd cycles, even cycles, color-critical graphs and intersecting triangles. The second goal is to conclude some recent results of the spectral conditions on some graphical properties. By …
