Let $G$ be a Cayley graph of the elementary abelian $2$-group $\mathbb{Z}_2^{n}$ with respect to a set $S$ of size $d$. We prove that for any such $G, S$ and $d$, the maximum degree of any induced subgraph of $G$ on any set of more than half the vertices is at least $\sqrt d$. This is deduced from the recent breakthrough result of Huang who proved the above for the $n$-hypercube $Q^n$, in which the set of generators $S$ is the set of all vectors of Hamming weight $1$. Motivated by his method we define and study unitary signings of adjacency matrices of graphs, and compare them to the orthogonal signings of Huang. As a byproduct, we answer a recent question of Belardo et. al. about the spectrum of signed $5$-regular graphs.
Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$, where $d$ is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian $2$-group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove thi…
In this paper, we propose the following conjecture which generalizes a theorem proved by Huang [Hua19] in his recent breakthrough proof of the sensitivity conjecture. We conjecture that for any Cayley graph $X = Γ(G,S)$ on a group $G$ and any generating set $S$, if $U \subseteq G$ has size $|U| > |G|/2$, then the induced subgraph of $X$ on $U$ has maximum degree at least $\sqrt{|S|/2}$. Using a recent idea of Alon and Zheng [AZ20], who proved this conjecture for the special case when $G = Z_2^n$, we prove that this conjecture is true whenever $G$ is abelian. We also observe that for this conje…
We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least $\log^c |G|$, for any fixed $c > 1$ and $|G|$ large enough.
In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $λ_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $λ_n=\frac{1+ε_n}{n-1}$ where $n^{-{1/3}+δ}\le ε_n<1$ and $δ>0$, a random induced subgraph has a.s. a unique largest component of size $\wp(ε_n)\frac{1+ε_n}{n-1}n!$, where $\wp(ε_n)$ is the survival probability of a specific branchi…
Let $Γ$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(Γ,T)$ be a Cayley graph of $Γ$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$ and we determine the second eigenvalues of a majority of the connected normal Cayley g…
This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that ${\rm{Aut}}(\Cay)$ is the product of the right translation group by $\textsf{N}\rtimes \textsf{D}_{n+1},$ where $\textsf{N}$ is the subgroup fixing $S_n$ element-wise and $\textsf{D}_{n+1}$ is a dihedral group of order $2(n+1)$. We conjecture that $\textsf{N}$ is trivial. We also prove that the subgraph $Γ$ with vertex-set $S_n$ is a $2(n-2)$-regular graph whose aut…
Let $Γ_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $Γ_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. The…
A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture.
We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. Specific results we prove include: (1) The signed join of a negative 2-clique with any positive (n,3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. (2) A signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for crea…
This paper deals with the Cayley graph $\mathrm{Cay}(\mathrm{Sym}_n,T_n),$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translation group by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $Γ$ of $\mathrm{Cay}(\mathrm{Sym}_n,T_n)$ induced by the set $T_n$. In particular, $Γ$ is a $2(n-2)$-regular graph whose automo…
Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a vector space over a finite field $Z/pZ$, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $clog nloglog n,$ where $c$ is an absolute constant. In this article we observe that a modification of his arguments shows that for an arbitrary finite abelian group of order $n$, there is a Cayley graph containing…
Given graphs $G$ and $H$, the generalized Turán number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of $G$. In this paper, we obtain an asymptotic upper bound on ${\rm ex}(CT_n,C_{2l})$ for any $n \ge 3$ and $l\geq2$, where $C_{2l}$ is the cycle of length $2l$ and $CT_n$ is the complete transposition graph which is defined as the Cayley graph on the symmetric group ${\rm S}_n$ with respect to the set of all transpositions of ${\rm S}_n$.
We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a spanning tree in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this subtree is a Hamiltonian cycle, but convergence is meant in a stronger sense. These latter are related to whether having a Hamiltonian cycle is a testable graph property.
A subset $R$ of the vertex set of a graph $Γ$ is said to be $(κ,τ)$-regular if $R$ induces a $κ$-regular subgraph and every vertex outside $R$ is adjacent to exactly $τ$ vertices in $R$. In particular, if $R$ is a $(κ,τ)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(κ,τ)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $κ$ and $τ$ a pair of integers satisfying $0\leqκ\leq|H|-1$, $1\leqτ\leq|H|$ and $\gcd(2,|H|-1)\midκ$. It is proved that (i) if $τ$ is even, then $H$ is a $(κ,τ)$-regular set of $G$; (ii) if $τ$ is odd, then $H$ is a $(κ…
Huang proved that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree $1$ on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner a…
Let us say that a Cayley graph $Γ$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $Γ$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$. In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of {Č}ern{ý} Cayley graphs.
Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $η$-closed if the average degree of the subgraph induced by $A$ is at least $ηd$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $η$. The work of this paper was motivated by an attempt to obtain a qualitative description of closed subsets of the Cayley graph $Γ$ whose vertex set is $\mathbb F_2^{n_1}\otimes \dots \otimes \mathbb F_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdot…
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
Determining the maximum number of edges in an $n$-vertex $C_4$-free graph is a well-studied problem that dates back to a paper of Erdős from 1938. One of the most important families of $C_4$-free graphs are the Erdős-Rényi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erdős-Rényi orthogonal polarity graph. Using this isomorphism we prove that the Petersen graph is a subgraph of every sufficiently large Erdős-Rényi orthogonal polarity graph.
Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\…
The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact value of $c(k, g)$ is known only for some small values of $k, g$ and three infinite families where $g \in \{6, 8, 12\}$ and $k - 1$ is a prime power. These infinite families come from the incidence graphs of generalized polygons. Some of the best known upper bounds on $c(k,g)$ for $g \in \{6, 8, 12\}$ have been obtained by constructing small regular induced subg…
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
Given a set $ξ=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $ξ$-free graph is one which does not contain any member of $% ξ$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let ${\gr{W}}_{k,ξ}$ be theexponential generating function (EGF for brief) of connected $ξ$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $ξ$, a fundamental differential recurrence satisfied by the EGFs ${\gr{W}}_{k,ξ}$ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of grap…
Disproof of the tree product conjecture via the Heisenberg group
Department of Mathematics, University College London（伦敦大学学院数学系）；Department of Mathematics and Statistics, McGill University（麦吉尔大学数学与统计系）；Department of Mathematics, ETH Zürich（苏黎世联邦理工学院数学系）
