Regular Uniform Hypergraphs, s-Cycles, s-Paths and Their largest Laplacian H-Eigenvalues∗

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph, where 1 ≤ s ≤ k−1, is regular if and only if there is a positive integer q such that k = q(k − s). We show that an even-uniform s-path and an even-uniform non-regular s-cycle are always oddbipartite. We prove that a regular s-cycle G with k = q(k − s) is odd-bipartite if and only if m is a multiple of 2t0 , where m is the number of edges in G, and q = 20(2l0 +1) for some integers t0 and l0. We identify the value of the largest signless Laplacian Heigenvalue of an s-cycle G in all possible cases. When G is odd-bipartite, this is also its largest Laplacian H-eigenvalue. We introduce supervertices for hypergraphs, and show the components of a Laplacian H-eigenvector of an odd-uniform hypergraph are equal if such components corresponds vertices in the same supervertex, and the corresponding Laplacian H-eigenvalue is not equal to the degree of the supervertex. Using this property, we show that the largest Laplacian H-eigenvalue of an odd-uniform generalized loose s-cycle G is equal to ∆(G) = 2. We also show that the largest Laplacian H-eigenvalue of a k-uniform tight s-cycle G is not less than ∆(G) + 1, if the number of edges is even and k = 4l + 3 for some nonnegative integer l.