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Let G be a simple graph with the adjacency matrix A (G) , and let τ (G) denote the smallest positive eigenvalue of A (G). Let G n be the class of all connected bipartite graphs on n = 2 k vertices with a unique perfect matching. In this article, we characterize the graphs G in G n such that τ (G) does not exceed 1 2. Using the above characterization, we obtain the unique graphs in G n with the maximum and the second maximum τ , respectively. Further, we prove that the largest and the second largest limit points of the smallest positive eigenvalues of bipartite graphs with a unique perfect matching are 1 2 and the reciprocal of α 3 1 2 + α 3 − 1 2 , respectively, where α 3 is the largest root of x 3 − x − 1. [ABSTRACT FROM AUTHOR]

A graph G is F -free, if it contains no F as a subgraph. A Brualdi-Solheid-Turán type problem asks what is the maximum spectral radius of an F -free graph of order n ? Let K a , b • K 3 denote the graph obtained by identifying a vertex of K a , b belonging to the part of size b and a vertex of K 3. In this paper, we consider about a Brualdi-Solheid-Turán type problem on non-bipartite graphs. We prove that if G is a non-bipartite graph with order n ≥ 21 and ρ (G) ≥ ρ (K ⌈ n − 2 2 ⌉ , ⌊ n − 2 2 ⌋ • K 3) , then G contains a pentagon unless G ≅ K ⌈ n − 2 2 ⌉ , ⌊ n − 2 2 ⌋ • K 3. [ABSTRACT FROM AUTHOR]

A graph G is called H -free, if it does not contain H as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of an H -free graph of order n ? In this paper, we consider the Brualdi-Solheid-Turán type problem for non-bipartite graphs. Let K a , b • K 3 denote the graph obtained by identifying a vertex of K a , b in the part of size b and a vertex of K 3. We prove that if G is a non-bipartite graph of order n satisfying ρ (G) ≥ ρ (K ⌈ n − 2 2 ⌉ , ⌊ n − 2 2 ⌋ • K 3) , then G contains all odd cycles C 2 l + 1 for each integer l ∈ [ 2 , k ] unless G ≅ K ⌈ n − 2 2 ⌉ , ⌊ n − 2 2 ⌋ • K 3 , provided that n is sufficiently large with respect to k. This resolves the problem posed by Guo, Lin and Zhao (2021). [ABSTRACT FROM AUTHOR]

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using the notion of partitioned tensor products. This extends the construction of Ji, Gong, and Wang [9]. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction [4] , and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al. [ABSTRACT FROM AUTHOR]