Uniform Hypertrees Research Articles

The Algebraic Multiplicity of the Spectral Radius of a Uniform Hypertree

It is well-known that the spectral radius of a connected uniform hypergraph is an eigenvalue of the hypergraph. However, its algebraic multiplicity remains unknown. In this paper, we use the Poisson Formula and the matching polynomials to give the algebraic multiplicity of the spectral radius of a uniform hypertree.

Rainbow connection numbers of some classes of $ s $-overlapping $ r $-uniform hypertrees with size $ t $

<abstract><p>The rainbow connection concept was developed to determine the minimum number of passwords required to exchange encrypted information between two agents. If the information exchange involves divisions managing more than two agents, the rainbow connection concept can be extended to a hypergraph. In 2014, Carpentier et al. expanded the rainbow connection concept of graphs to hypergraphs. They implemented it on a minimally connected hypergraph, an $ r $-uniform complete hypergraph, an $ r $-uniform cycle hypergraph, and an $ r $-uniform complete multipartite hypergraph. However, they did not determine the rainbow connection numbers of hypertrees. A hypergraph $ \mathcal{H} $ is called a hypertree if there exists a host tree $ T $ such that each edge of $ \mathcal{H} $ induces a subtree in $ T $. Therefore, in this article, we consider the rainbow connection numbers of some classes of $ s $-overlapping $ r $-uniform hypertrees with size $ t $. For $ r\geq 2 $, $ 1\leq s < r $, and $ t\geq 1 $, an $ s $-overlapping $ r $-uniform hypertree with size $ t $ is an $ r $-uniform connected hypertree, with $ s $ being the maximum cardinality of the vertex set obtained from the intersection of each pair of edges. We provide the best lower bound of the rainbow connection number of a connected hypergraph. Then, we determine the rainbow connection numbers of six classes of $ s $-overlapping $ r $-uniform hypertrees with size $ t $.</p></abstract>

The hypertree with the second largest spectral radius among uniform hypertrees with given size and stability number

The hypertree with the second largest spectral radius among uniform hypertrees with given size and stability number

The Hypertree with the Largest Spectral Radius Among all Hypertrees with Given Size and Stability Number

In this paper, we determine the unique uniform hypertree with the maximum spectral radius in the set (denoted by $${\cal T}(m,r,\alpha)$$ ) of all r-uniform hypertrees with given size m(> r) and stability number α. More precisely, for any $$T \in {\cal T}$$ with m > r, we have $$\rho (T) \le {\left({{1 \over {1 - {\alpha _0}}}} \right)^{1/r}},$$ where α0 is the maximum root of $${x^{r - 1}}\left({{1 \over {1 - x}} - {1 \over {{x^s}}} - l} \right) = g,$$ and integers g, l, s are determined by n − 1 − α = g(r − 1) + s, where n = m(r − 1) + 1 is the order of T and 0 ≤ s < r − 1, and l = m − gr − s − 1. Further, $$\rho (T) = {({1 \over {1 - {\alpha _0}}})^{1/r}}$$ if and only if T = R((r − 1)g; 0l; s).

Distance (Signless) Laplacian Eigenvalues of $k$-uniform Hypergraphs

The distance (signless) Laplacian eigenvalues of a connected hypergraph are the eigenvalues of its distance (signless) Laplacian matrix. For all $n$-vertex $k$-uniform hypertrees, we determine the $k$-uniform hypertree with minimum second largest distance (signless) Laplacian eigenvalue. For all $n$-vertex $k$-uniform unicyclic hypergraphs, we obtain the $k$-uniform unicyclic hypergraph with minimum largest distance (signless) Laplacian eigenvalue, and the $k$-uniform unicyclic hypergraph with minimum second largest distance Laplacian eigenvalue.

Decomposing uniform hypergraphs into uniform hypertrees and single edges

Given two r-uniform hypergraphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a hypergraph isomorphic to H. Let ϕr(n,H) be the smallest integer such that any r-uniform hypergraph G of order n admits an H-decomposition with at most ϕr(n,H) parts. In this paper we determine the exact value of ϕr(n,H) when H is an arbitrary r-uniform hypertree with t edges.

Graceful Labeling of Hypertrees

Graph labeling is considered as one of the most interesting areas in graph theory. A labeling for a simple graph G (numbering or valuation), is an association of non -negative integers to vertices of G&amp;nbsp; (vertex labeling) or to edges of G&amp;nbsp; (edge labeling) or both of them. In this paper we study the graceful labeling for the k- uniform hypertree and define a condition for the corresponding tree to be graceful. A k- uniform hypertree is graceful if the minimum difference of vertices&amp;rsquo; labels of each edge is distinct and each one is the label of the corresponding edge.

The second largest spectral radii of uniform hypertrees with given size of matching

ABSTRACT In this paper, we almost completely determine the hypertrees with the second largest spectral radius among all hypertrees with m edges and given size of matching by using the matching polynomials and ordering of hypertrees.

The Fifth Maximum Wiener Index of Uniform Hypergraphs

Hypergraph theory has been found many applications in chemistry. As an important descriptor of molecular structures, the Wiener index of a graph also has many applications. The Wiener index of a connected hypergraph is defined as the summation of distances between all pairs of vertices. If each edge contains exactly k vertices, then a hypergraph G is called k-uniform. A hypertree is a connected hypergraph with no cycles. For k-uniform hypertree, H. Guo, B. Zhou et al. have determined the first, second and third maximum and minimum Wiener indices of uniform hypertrees. And give the unique structure of the k-uniform hypertree corresponding to the Wiener index, Moreover, in this paper, We first find out the relationship between the first few Wiener indices, then according to the structure of the graph, determine the unique k-uniform hypertree with the fifth maximum Wiener index. Through the determination of the fifth Wienr index k-uniform hypertree, the structure of the NTH Wiener index k-uniform hypertree can be found.

The largest spectral radius of uniform hypertrees with a given size of matching

ABSTRACT In this paper we consider the spectral radii of a special set system called hypertrees, i.e. connected acyclic hypergraphs. By employing the theory of matching polynomial of hypertrees and ordering of hypertrees, we determine the largest spectral radius of hypertrees with m edges and given size of matching.

On the spectral radius of uniform hypertrees

We determine the unique hypertrees with maximum spectral radius among k-uniform non-hyper-caterpillars with m edges and diameter d for 4≤d≤m−2, among k-uniform non-hyper-caterpillars with m edges for m≥6, among k-uniform hypertrees with m edges and independence number α for ⌈m(k−1)+1k⌉≤α≤m, among k-uniform hypertrees with m edges and matching number β for 1≤β≤⌊m(k−1)+1k⌋, and among k-uniform hypertrees with m edges and s branch vertices for 0≤s≤⌊m−12⌋, respectively.

On the Adjacency Spectra of Hypertrees

We show that $\lambda$ is an eigenvalue of a $k$-uniform hypertree $(k \geq 3)$ if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to provide a spectral characterization for power hypertrees. Notably, the situation is quite different from that of ordinary trees, i.e., $2$-uniform trees. We conclude by presenting an example (an $11$ vertex, $3$-uniform non-power hypertree) illustrating these phenomena.

The spectra of uniform hypertrees

In this paper we study the spectra of uniform hypertrees by using the generalized weighted incident matrix. We show that λ is a nonzero eigenvalue of the hypertree H corresponding to an eigenvector with all elements nonzero if and only if λ is a root of the polynomial φ(H)=∑i=0m(−1)i|Mi|x(m−i)r, where |Mi| is the number of matchings of order i in H.

Plurigraph Coloring and Scheduling Problems

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contraction law can be applied to scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exists pairs of 3-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.

Bounds on the number of edges of edge-minimal, edge-maximal and l-hypertrees

In their paper, Bounds on the Number of Edges in Hypertrees, G.Y. Katona and P.G.N. Szab\'o introduced a new, natural definition of hypertrees in $k$-uniform hypergraphs and gave lower and upper bounds on the number of edges. They also defined edge-minimal, edge-maximal and $l$-hypertrees and proved an upper bound on the edge number of $l$-hypertrees. In the present paper, we verify the asymptotic sharpness of the $\binom{n}{k-1}$ upper bound on the number of edges of $k$-uniform hypertrees given in the above mentioned paper. We also make an improvement on the upper bound of the edge number of $2$-hypertrees and give a general extension construction with its consequences. We give lower and upper bounds on the maximal number of edges of $k$-uniform edge-minimal hypertrees and a lower bound on the number of edges of $k$-uniform edge-maximal hypertrees. In the former case, the sharp upper bound is conjectured to be asymptotically $\frac{1}{k-1}\binom{n}{2}$.

Trees in greedy colorings of hypergraphs

It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The ‘standard’ reasoning is that such a graph must contain a subgraph of minimum degree at least t . Bohman, Frieze, and Mubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh recently proved that every uniform hypergraph of chromatic number larger than t contains every hypertree with t edges. Here we observe that the basic property of the well-known greedy algorithm immediately implies a much more general result (with a conceptually simpler proof): if the greedy algorithm colors the vertices of an r -uniform hypergraph with more than t colors then the hypergraph contains every r -uniform hypertree with t edges.

On chromaticity of hypergraphs

Let H = ( X , E ) be a simple hypergraph and let f ( H , λ ) denote its chromatic polynomial. Two hypergraphs H 1 and H 2 are chromatic equivalent if f ( H 1 , λ ) = f ( H 2 , λ ) . The equivalence class of H is denoted by 〈 H 〉 . Let K and H be two classes of hypergraphs. H is said to be chromatically characterized in K if for every H ∈ H ∩ K we have 〈 H 〉 ∩ K = H ∩ K . In this paper we prove that uniform hypertrees and uniform unicyclic hypergraphs are chromatically characterized in the class of linear hypergraphs.