According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer $m$ and group $G$ (finite or infinite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set $Ω$, whose automorphism group is the corresponding diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups.
Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal graph $Γ_D(G,m)$ is a Cayley graph for the group~$G^m$, and so is vertex-transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the Möbius function of the diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph.
We believe that this family of graphs will play a significant role in algebraic graph theory.
Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups,…
We attach to each $\langle 0, \vee \rangle$-semilattice a graph $\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency relation. We study properties of the graph $\boldsymbol{G}_{\boldsymbol{S}}$ both when $\boldsymbol{S}$ is a join-semilattice and when it is a lattice. We call a $\langle 0, \vee \rangle$-semilattice $\boldsymbol{S}$ particle provided that the set of its join-irreducible elements join-generates $\boldsymbol{S}$ and it satisfies DCC. We prove that the congruence lattice of a parti…
In this paper, we introduce the graph $G(S)$ of a bounded semilattice $S$, which is a generalization of the intersection graph of the substructures of an algebraic structure. We prove some general theorems about these graphs; as an example, we show that if $S$ is a product of three or more chains, then $G(S)$ is Eulerian if and only if either the length of every chain is even or all the chains are of length one. We also show that if $G(S)$ contains a cycle, then $girth(G(S)) = 3$. Finally, we show that if $(S,+,\cdot,0,1)$ is a dually atomic bounded distributive lattice whose set of dual atoms…
In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $m\ge2$, a set of $m+1$ partitions of a set $Ω$, any $m$ of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $m=2$), or generate a join-semilattice of dimension $m$ associated with a diagonal group over a base group $G$. In this paper we investigate what happens if we have $m+r$ partitions with $r\geq 2$, any $m$ of which are minimal elements of a Cartesian lattice. If $m=2$, this is just a set of mutually orthogonal Latin squares. We consider the cas…
We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.
Directed graphs have long been used to gain understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Fruct's Theorem. We investigate four inverse semigroups defined over undirected graphs constructed from the notions of subgraph, vertex induced subgraph, rooted tree induced subgraph, and rooted path induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these four inverse semigroups. Finally, we prove a characterization …
Classical Hamming graphs are Cartesian products of complete graphs, and two vertices are adjacent if they differ in exactly one coordinate. Motivated by connections to unitary Cayley graphs, we consider a generalization where two vertices are adjacent if they have no coordinate in common. Metric dimension of classical Hamming graphs is known asymptotically, but, even in the case of hypercubes, few exact values have been found. In contrast, we determine the metric dimension for the entire diagonal family of $3$-dimensional generalized Hamming graphs. Our approach is constructive and made possib…
The purpose of this paper is to establish a new continuous-time on-diagonal lower estimate of heat kernel for large time on graphs. To achieve the goal, we first give an upper bound of heat kernel in natural graph metric, and then use this bound and the volume growth condition to show the validity of the on-diagonal lower bound.
In this paper, we obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=α$ over the finite field $\mathbb{F}_{p^{ab}}$, with $k=\frac{p^{ab}-1}{b(p^a-1)}$ and $b>1$ by using the number of $r$-walks in NEPS of complete graphs.
We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3) prove that the icosahedral graph is diagonal, and (4) compute the magnitude homology of cycles. These results answer several questions of Hepworth and Willerton [HW17].
We study the diagonal entries of the average mixing matrix of continuous quantum walks. The average mixing matrix is a graph invariant; it is the sum of the Schur squares of spectral idempotents of the Hamiltonian. It is non-negative, doubly stochastic and positive semi-definite. We investigate the diagonal entries of this matrix. We study the graphs for which the trace of the average mixing matrix is maximum or minimum and we classify those which are maximum. We give two constructions of graphs whose average mixing matrices have constant diagonal.
Let the star on $n$ vertices, namely $K_{1,n-1}$ be denoted by $S_n$. If every two coloring of the edges of a complete balanced multipartite graph $K_{j \times s}$ there is a copy of $S_n$ in the first color or a copy of $S_m$ in the second color, then we will say $K_{j \times s}\rightarrow (S_n,S_m)$. The size Ramsey multipartite number $m_j(S_n, S_m)$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (S_n,S_m)$. In this paper, we obtain the exact values of the size Ramsey numbers $m_j(S_n,S_m)$ for $n,m \ge 3$ and $j \ge 3$.
A nuciferous graph is a simple graph with a non-singular $0$-$1$ adjacency matrix $A$ such that all the diagonal entries of $A^{-1}$ are zero and all the off-diagonal entries of $A^{-1}$ are non-zero. Sciriha et al. conjectured that except $K_2$, no nuciferous graph exists. We disprove this conjecture. Moreover, we conjecture that there infinitely many nuciferous Cayley graphs.
Trotter and Erdös found conditions for when a directed $m \times n$ grid graph on a torus is Hamiltonian. We consider the analogous graphs on a two-holed torus, and study their Hamiltonicity. We find an $\mathcal{O}(n^4)$ algorithm to determine the Hamiltonicity of one of these graphs and an $\mathcal{O}(\log(n))$ algorithm to find the number of diagonals, which are sets of vertices that force the directions of edges in any Hamiltonian cycle. We also show that there is a periodicity pattern in the graphs' Hamiltonicities if one of the sides of the grid is fixed; and we completely classify whic…
A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n, 1, \dots, 1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades and Sagan about the $(q, q^{-1})$-Ehrhart fun…
Finding a diagonal matrix congruent to $A - cI$ for constants $c$, where $A$ is the adjacency matrix of a graph $G$ allows us to quickly tell the number of eigenvalues in a given interval. If $G$ has clique-width $k$ and a corresponding $k$-expression is known, then diagonalization can be done in time $O(\text{poly}(k) n)$ where $n$ is the order of $G$.
Let $G$ be a simple graph on the vertex set $\{1,\ldots,n\}$ with $m$ edges. An algebraic object attached to $G$ is the ideal $P_{G}$ generated by diagonal 2-minors of an $n \times n$ matrix of variables. In this paper we prove that if $G$ is bipartite, then every initial ideal of $P_{G}$ is generated by squarefree monomials of degree at most $\left \lfloor{\frac{m+n+1}{2}} \right \rfloor$. Furthermore, we completely characterize all connected graphs $G$ for which $P_{G}$ is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Gr{ö}bner basis o…
We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents a natural barrier to improvement.
Let $0 \in Γ$ and $Γ\setminus \{0\}$ be a subgroup of the complex numbers of unit modulus. Define $\mathcal{H}_{n}(Γ)$ to be the set of all $n\times n$ Hermitian matrices with entries in $Γ$, whose diagonal entries are zero. The matrices $A,B\in \mathcal{H}_{n}(Γ)$ are said to be switching equivalent if there is a diagonal matrix $D$, in which the diagonal entries belong to $Γ\setminus \{0\}$, such that $D^{-1} A D=B$. We find a characterization, in terms of fundamental cycles of graphs, of switching equivalence of matrices in $\mathcal{H}_{n}(Γ)$. We give sufficient conditions to characterize…
For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the "off-diagonal" ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the m…
For two graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the ordered Ramsey number $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph on $N$ vertices contains a red copy of $G^<$ or a blue copy of $H^<$. For a positive integer $n$, a nested matching $NM^<_n$ is the ordered graph on $2n$ vertices with edges $\{i,2n-i+1\}$ for every $i=1,\dots,n$. We improve bounds on the ordered Ramsey numbers $r_<(NM^<_n,K^<_3)$ obtained by Rohatgi, we disprove his conjecture by showing $4n+1 \leq r_<(NM^<_n,K^<_3) \leq (3+\sqrt{5})n$ for eve…
For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a polynomial in $n$ algorithm deciding whether $R(M) \leq k$ (whose complexity may depend on $k$). We also give a polynomial in $n$ algorithm computing a number $m$ such that $m/2 \leq R(M) \leq m$. These results have applications to graph drawings on non-orientable surfaces.
