序列$S$是潜在的$K_{m}-C_{4}$ -图形的，如果它有一个实现包含一个$K_{m}-C_{4}$作为子图。令$σ(K_{m}-C_{4}, n)$表示最小度和使得每个具有$σ(S)\geq σ(K_{m}-C_{4}, n)$的$n$项图序列$S$是潜在$K_{m}-C_{4}$ -图的。本文证明了$σ(K_{m}-C_{4}, n)\geq (2m-6)n-(m-3)(m-2)+2,$对$n \geq m \geq 4.$成立。我们猜想等式对$n \geq m \geq 4.$成立。我们证明了这个猜想对$m=5$成立。
英文摘要：
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 $m=5$.
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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 …
