Let H =(V, E) be a k-uniform hypergraph. For 1 ≤ s ≤ k-1, an s-path P^(k,s)_n of length n in H is a sequence of distinct vertices v_1, v_2, · · ·, v_(s+n(k-s)) such that {v_(1+i(k-s)), · · &...Let H =(V, E) be a k-uniform hypergraph. For 1 ≤ s ≤ k-1, an s-path P^(k,s)_n of length n in H is a sequence of distinct vertices v_1, v_2, · · ·, v_(s+n(k-s)) such that {v_(1+i(k-s)), · · ·, v_(s+(i+1)(k-s))} is an edge of H for each 0 ≤ i ≤ n-1.In this paper, we prove that R(P^(3 s,s)_n, P^(3 s,s)_3) =(2 n + 1)s + 1 for n ≥ 3.展开更多
Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs ar...Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.展开更多
We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3,...We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3, we show that the largest H-eigenvalue of its adjacency tensor is （（1 ＋ √-5）/2）2/k when = 3 and ）λ（A） = 31/k when g = 4, respectively. For the case of l ≥ 5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval （2, 3） when l≥ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.展开更多
文摘Let H =(V, E) be a k-uniform hypergraph. For 1 ≤ s ≤ k-1, an s-path P^(k,s)_n of length n in H is a sequence of distinct vertices v_1, v_2, · · ·, v_(s+n(k-s)) such that {v_(1+i(k-s)), · · ·, v_(s+(i+1)(k-s))} is an edge of H for each 0 ≤ i ≤ n-1.In this paper, we prove that R(P^(3 s,s)_n, P^(3 s,s)_3) =(2 n + 1)s + 1 for n ≥ 3.
文摘Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271221） and the Specialized Research Fund for State Key Laboratories.
文摘We investigate k-uniform loose paths. We show that the largest H- eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l≥ 3, we show that the largest H-eigenvalue of its adjacency tensor is （（1 ＋ √-5）/2）2/k when = 3 and ）λ（A） = 31/k when g = 4, respectively. For the case of l ≥ 5, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval （2, 3） when l≥ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.
