Spectral bipartite Turán problems on linear hypergraphs

Let F be a graph, and let $B_r\left(F\right)$ be the class of r-uniform Berge-F hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for $B_r\left(C_3\right)$-free linear r-uniform hypergraphs and upper bounds for the spectral radii of $B_r\left(K_{2,t}\right)$-free or $\left\{B_r\right(K_{s,t}),B_r(C_3\left)\right\}$-free linear r-uniform hypergraphs, where $C_3$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having s vertices and the other part having t vertices. Our work implies an upper bound for the number of edges of $\left\{B_r\right(K_{s,t}),B_r(C_3\left)\right\}$-free linear r-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.