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⩾3k⩾3, reaches its upper bound 2Δ(G)2Δ(G), where Δ(G)Δ(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 odd-bipartite. We show that an s-cycle G, as a k  -uniform hypergraph, where 1⩽s⩽k−11⩽s⩽k−1, is regular if and only if there is a positive integer q   such that k=q(k−s)k=q(k−s). We show that an even-uniform s-path and an even-uniform non-regular s-cycle are always odd-bipartite. We prove that a regular s-cycle G   with k=q(k−s)k=q(k−s) is odd-bipartite if and only if m   is a multiple of 2t02t0, where m is the number of edges in G  , and q=2t0(2l0+1)q=2t0(2l0+1) for some integers t0t0 and l0l0. We identify the value of the largest signless Laplacian H-eigenvalue 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 correspond 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 Δ(G)=2Δ(G)=2. We also show that the largest Laplacian H-eigenvalue of a k-uniform tight s-cycle G   is not less than Δ(G)+1Δ(G)+1, if the number of vertices is even and k=4l+3k=4l+3 for some nonnegative integer l.