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Let n ≥ 29 , k ≥ 2 and δ ≥ 1 be three positive integers, where n ≥ k + 2 and n ≡ k (mod 2). In this paper, we characterize the extremal graphs with maximum spectral radius among all connected graphs on n vertices with minimum degree δ and matching number at most n − k 2. This solves an open problem proposed in [10]. [ABSTRACT FROM AUTHOR]

For a graph G and an integer k ≥ 2 , the k-power of G is the graph with vertex set V (G) in which two vertices are adjacent if and only if their distance is at most k in G. In this paper, we study spectral conditions for a graph to contain a copy of k -power of Hamilton cycle. This solves a problem proposed in [20]. [ABSTRACT FROM AUTHOR]

A graph G is called a 1-planar graph if G has a drawing in the plane, so that each of its edges is crossed at most once. In this paper, we prove that λ 1 (G) ≤ 5 + 2 n + 5 for a 1-planar graph G of order n ≥ 7. For n ≥ 6.23 × 10 18 , we characterize the unique extremal graph with the maximum spectral radius among all 1-planar graphs of order n. [ABSTRACT FROM AUTHOR]

A matching of a graph is a set of disjoint edges. Let n , t and k be three positive integers, where k ≥ 2 , n ≥ k + 2 , t ≥ 1 and n ≡ k (mod 2). In this paper, we characterize the extremal graphs with maximum spectral radius among all t -connected graphs on n vertices with matching number at most n − k 2. This generalizes a result of O (2021) [12]. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices with m edges. Let d 1 , d 2 , ... , d n be the degrees of the vertices in G. Define the degree deviation in G by s (G) = ∑ 1 ≤ i ≤ n | d i − 2 m n |. The spectral radius of G is the largest eigenvalue of the adjacent matrix of G , which is denoted by λ 1 (G). In this paper, we prove that λ 1 (G) − 2 m n ≤ 9 10 s (G) , which improves a result in [12]. Moreover, if G is bipartite, we prove that λ 1 (G) − 2 m n ≤ 1 2 s (G). This bound is tight for considering the star graph K 1 , n − 1. [ABSTRACT FROM AUTHOR]