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Register a new userThe Poset of Hypergraph Quasirandomness
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Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990s.
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The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set ${1,2,dots, n}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the Mobius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitneys NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.
The abstract induced subgraph poset of a graph is the isomorphism class of the induced subgraph poset of the graph, suitably weighted by subgraph counting numbers. The abstract bond lattice and the abstract edge-subgraph poset are defined similarly by considering the lattice of subgraphs induced by connected partitions and the poset of edge-subgraphs, respectively. Continuing our development of graph reconstruction theory on these structures, we show that if a graph has no isolated vertices, then its abstract bond lattice and the abstract induced subgraph poset can be constructed from the abstract edge-subgraph poset except for the families of graphs that we characterise. The construction of the abstract induced subgraph poset from the abstract edge-subgraph poset generalises a well known result in reconstruction theory that states that the vertex deck of a graph with at least 4 edges and without isolated vertices can be constructed from its edge deck.12
The Stern poset $mathcal{S}$ is a graded infinite poset naturally associated to Sterns triangle, which was defined by Stanley analogously to Pascals triangle. Let $P_n$ denote the interval of $mathcal{S}$ from the unique element of row $0$ of Sterns triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $ngeq 1$ let begin{align*} L_n(q)&=2cdotleft(sum_{k=1}^{2^n-1}A_{P_k}(q)right)+A_{P_{2^n}}(q), end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $ngeq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(q)$ is divisible by $L_{2n}(q)$. In this paper we obtain a simple recurrence relation satisfied by $L_n(q)$ and affirmatively solve Stanleys conjectures. We also establish the asymptotic normality of the coefficients of $L_n(q)$.
Let $H$ be connected $m$-uniform hypergraph and $mathcal{A}(H)$ be the adjacency tensor of $H$. The stabilizing index of $H$, denoted by $s(H)$, is exactly the number of eigenvectors of $mathcal{A}(H)$ associated with the spectral radius, and the cyclic index of $H$, denoted by $c(H)$, is the number of eigenvalues of $mathcal{A}(H)$ with modulus equal to the spectral radius. Let $bar{H}$ be a $k$-fold covering of $H$. Then $bar{H}$ is isomorphic to a hypergraph $H_B^phi$ derived from the incidence graph $B_H$ of $H$ together with a permutation voltage assignment $phi$ in the symmetric group $mathbb{S}_k$. In this paper, we first characterize the connectedness of $bar{H}$ by using $H_B^phi$ for subsequent discussion. By applying the theory of module and group representation, we prove that if $bar{H}$ is connected, then $s(H) mid s(bar{H})$ and $c(H) mid c(bar{H})$. In the situation that $bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $mathcal{A}(bar{H})$ contains the spectrum of $mathcal{A}(H)$ and the spectrum of a signed hypergraph constructed from $H$ and the covering projection; if $m$ is odd, we give an explicit formula for $s(bar{H})$.
Let H be a hypergraph on n vertices with the property that no edge contains another. We prove some results for a special case of the Isolation Lemma when the label set for the edges of H can only take two values. Given any set of vertices S and an edge e, the weight of S in e is the size of e plus the size of the intersection of S and e. A versal S for an edge e is a set of vertices with weight in e smaller than the weight in any other edge. We show that H always has at least n + 1 versals except if H is either the set of all singletons T_n or the complement of T_n or the 4-cycle graph. In those exceptional cases there are only n versals.
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