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Let $F$ be a graph. A hypergraph is called Berge $F$ if it can be obtained by replacing each edge in $F$ by a hyperedge containing it. Given a family of graphs $mathcal{F}$, we say that a hypergraph $H$ is Berge $mathcal{F}$-free if for every $F in mathcal{F}$, the hypergraph $H$ does not contain a Berge $F$ as a subhypergraph. In this paper we investigate the connections between spectral radius of the adjacency tensor and structural properties of a linear hypergraph. In particular, we obtain a spectral version of Tur{a}n-type problems over linear $k$-uniform hypergraphs by using spectral methods, including a tight result on Berge $C_4$-free linear $3$-uniform hypergraphs.
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In this paper, we characterize the extremal digraphs with the maximal or minimal $alpha$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph classes are denoted by $mathcal{R}_{m}^k$, $widetilde{boldsymbol{Theta}}_k(m)$ and $INF(m)$ respectively. The main results about spectral extremal digraph by Guo and Liu in cite{MR2954483} and Li and Wang in cite{MR3777498} are generalized to $alpha$-spectral graph theory. As a by-product of our main results, an open problem in cite{MR3777498} is answered. Furthermore, we determine the digraphs with the first three minimal $alpha$-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal $alpha$-spectral radius among all strongly connected digraphs for $0le alpha le frac{1}{2}$.
For every positive integer $t$ we construct a finite family of triple systems ${mathcal M}_t$, determine its Tur{a}n number, and show that there are $t$ extremal ${mathcal M}_t$-free configurations that are far from each other in edit-distance. We also prove a strong stability theorem: every ${mathcal M}_t$-free triple system whose size is close to the maximum size is a subgraph of one of these $t$ extremal configurations after removing a small proportion of vertices. This is the first stability theorem for a hypergraph problem with an arbitrary (finite) number of extremal configurations. Moreover, the extremal hypergraphs have very different shadow sizes (unlike the case of the famous Turan tetrahedron conjecture). Hence a corollary of our main result is that the boundary of the feasible region of ${mathcal M}_t$ has exactly $t$ global maxima.
We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].
We present a variant of a universality result of Rodl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the signless Laplacian of a graph and also suggests a natural way to define the adjacency and Laplacian matrices of a hypergraph. Some results presented generalize both graph and signed graph results to a hypergraphic setting.
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