This survey is two-fold. We first report new progress on the spectral extremal results on the
Tur\'{a} n type problems in graph theory. More precisely, we shall summarize the spectral …

Minors play a key role in graph theory, and extremal problems on forbidding minors have
attracted appreciable amount of interest in the past decades. In this paper, we focus on …

Turán-type problem is one of central problems in extremal graph theory. Erdős et al.[J.
Combin. Theory Ser. B 64 (1995) 89-100] obtained the exact Turán number of the friendship …

The gem is the 5-vertex graph consisting of a 4-vertex path plus a vertex adjacent to each
vertex of the path. A graph is said to be gem-free if it does not contain gem as a subgraph. In …

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and
eigenvectors of matrices associated with graphs to study them. In this paper, we present a …

In this paper, we give spectral conditions to guarantee the existence of two edge disjoint
cycles and two cycles of the same length. These two results can be seen as spectral …

Let F k be the (friendship) graph obtained from k triangles by sharing a common vertex. The
F k-free graphs of order n which attain the maximal spectral radius was firstly characterized …

Motivated by the counting results for color-critical subgraphs by Mubayi (2010), we study the
phenomenon behind Mubayi's theorem from a spectral perspective and start up this problem …

In the 1970s, Erdős asked how many edges are needed in a graph on $ n $ vertices, to
ensure the existence of a cycle of length exactly $ nk $. In this paper, we consider the …

Let ρ (G) be the spectral radius of a graph G with m edges. Nosal's theorem states that if ρ
(G)> m then G contains a triangle. In this article, we give a generalization: any non-bipartite …