Combinatorics (math.CO)

Abstract: The study of regular incidence structures such as projective planes and symmetric block designs is a well established topic in discrete mathematics. Work of Bruck, Ryser and Chowla in the mid-twentieth century applied the Hasse-Minkowski local-global theory for quadratic forms to derive non-existence results for certain design parameters. Several combinatorialists have provided alternative proofs of this result, replacing conceptual arguments with algorithmic ones. In this paper, we show that the methods required are purely linear-algebraic in nature and are no more difficult conceptually than the theory of the Jordan Canonical Form. Computationally, they are rather easier. We conclude with some classical and recent applications to design theory, including a novel application to the decomposition of incidence matrices of symmetric designs.

Abstract: Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of $L(X)\times K_m$ is equal to $m-1$, where $\times$ denotes the Kronecker product. We provide a few necessary and sufficient conditions for $L(X)\times K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)\times K_m$, where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.

Abstract: Let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices which contains no independent sets larger than $s$. The behaviour of $\mathrm{ex}(n,s)$ was first studied by Andr\'asfai, who conjectured that for $s>n/3$ this function is determined by appropriately chosen blow-ups of so called Andr\'asfai graphs. Moreover, he proved $\mathrm{ex}(n, s)=n^2-4ns+5s^2$ for $s/n\in [2/5, 1/2]$ and in earlier work we obtained $\mathrm{ex}(n, s)=3n^2-15ns+20s^2$ for $s/n\in [3/8, 2/5]$. Here we make the next step in the quest to settle Andr\'asfai's conjecture by proving $\mathrm{ex}(n, s)=6n^2-32ns+44s^2$ for $s/n\in [4/11, 3/8]$.

Abstract: The induced subgraph of the corresponding Grassmann graph formed by simplex codes is considered. We show that this graph, as the Grassmann graph, contains two types of maximal cliques. For any two cliques of the first type there is a monomial linear automorphism transferring one of them to the other. Cliques of the second type are more complicated and can contain different numbers of elements.

Abstract: PED partitions are partitions with even parts distinct while odd parts are unrestricted. Similarly, POD partitions have distinct odd parts while even parts are unrestricted. Merca proved several recurrence relations analytically for the number of PED partitions of $n$. They are similar to the recurrence relation for the number of partitions of $n$ given by Euler's pentagonal number theorem. We provide combinatorial proofs for all of these theorems and also for the pentagonal number theorem for PED partitions proved analytically by Fink, Guy, and Krusemeyer. Moreover, we prove combinatorially a recurrence for POD partitions given by Ballantine and Merca, Beck-type identities involving PED and POD partitions, and several other results about PED and POD partitions.

Abstract: Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and study sets of partitions in which parts with fixed residue(s) modulo r are distinct while all other parts are unrestricted. We also study partitions in which parts divisible by r (respectively congruent to r modulo 2r) must occur with multiplicity greater than one.

Abstract: A network $N$ on a finite set $X$, $|X|\geq 2$, is a connected directed acyclic graph with leaf set $X$ in which every root in $N$ has outdegree at least 2 and no vertex in $N$ has indegree and outdegree equal to 1; $N$ is arboreal if the underlying unrooted, undirected graph of $N$ is a tree. Networks are of interest in evolutionary biology since they are used, for example, to represent the evolutionary history of a set $X$ of species whose ancestors have exchanged genes in the past. For $M$ some arbitrary set of symbols, $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if there exists some arboreal network $N$ whose vertices with outdegree two or more are labelled by elements in $M$ and so that $d(\{x,y\})$, $\{x,y\} \in {X \choose 2}$, is equal to the label of the least common ancestor of $x$ and $y$ in $N$ if this exists and $\odot$ else. Important examples of symbolic arboreal maps include the symbolic ultrametrics, which arise in areas such as game theory, phylogenetics and cograph theory. In this paper we show that a map $d:{X \choose 2} \to M \cup \{\odot\}$ is a symbolic arboreal map if and only if $d$ satisfies certain 3- and 4-point conditions and the graph with vertex set $X$ and edge set consisting of those pairs $\{x,y\} \in {X \choose 2}$ with $d(\{x,y\}) \neq \odot$ is Ptolemaic. To do this, we introduce and prove a key theorem concerning the shared ancestry graph for a network $N$ on $X$, where this is the graph with vertex set $X$ and edge set consisting of those $\{x,y\} \in {X \choose 2}$ such that $x$ and $y$ share a common ancestor in $N$. In particular, we show that for any connected graph $G$ with vertex set $X$ and edge clique cover $K$ in which there are no two distinct sets in $K$ with one a subset of the other, there is some network with $|K|$ roots and leaf set $X$ whose shared ancestry graph is $G$.

Abstract: First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation.

Abstract: Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs $\mathcal{U}$, the tropicalization of the graph profile of $\mathcal{U}$ essentially records all valid pure binomial inequalities involving graph homomorphism numbers for graphs in $\mathcal{U}$. Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for $K_1$ and $S_{2,1^k}$-trees, almost-star graphs with one branch containing two edges and $k$ branches containing one edge. This allows pure binomial inequalities in homomorphism numbers (or densities) for these graphs to be verified through an explicit linear program where the number of variables is equal to the number of edges in the biggest $S_{2,1^k}$-tree involved.