In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is determined. A formula for the infection number for interval hypergraphs and several families of cyclic hypergraphs is given. The value of the infection number for a hypergraph whose edges form a symmetric t-design is given, and bounds are determined for a hypergraph whose edges are a t-design. Finally, the infection number for several hypergraph products and line graphs are considered.

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We determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order n ≥ 4, and thus we answer affirmatively the problem whether the unique digraph which achieves the minimum spectral radius among all strongly connected bicyclic digraphs of order n achieves the second minimum spectral radius among all strongly connected digraphs of order n for n ≥ 4 proposed in [H. Lin, J. Shu, A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524-2530. We also discuss the strongly connected bicyclic digraphs with small and large spectral radii respectively.

a b s t r a c tLet D(G) = (d i,j ) n×n denote the distance matrix of a connected graph G with order n, where d ij is equal to the distance between v i and v j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ϱ(G). In this paper, we give some graft transformations that decrease and increase ϱ(G) and prove that the graph S ′ n (obtained from the star S n on n (n is not equal to 4, 5) vertices by adding an edge connecting two pendent vertices) has minimal distance spectral radius among unicyclic graphs on n vertices; while P ′ n (obtained from a triangle K 3 by attaching pendent path P n−3 to one of its vertices) has maximal distance spectral radius among unicyclic graphs on n vertices.