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Sanming Zhou

Publications and source records attributed to Sanming Zhou.

At least 37 records · Page 2Linked to original sources
arXiv2023

The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles

We study the normal Cayley graphs $\mathrm{Cay}(S_n, C(n,I))$ on the symmetric group $S_n$, where $I\subseteq \{2,3,\ldots,n\}$ and $C(n,I)$ is the set of all cycles in $S_n$ with length in $I$. We prove that the strictly second largest eigenvalue of $\mathrm{Cay}(S_n,C(n,I))$ can only be achieved by at most four irreducible representations of $S_n$, and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when $I$ contains neither $n-1$ nor $n$ we know exactly when $\mathrm{Cay}(S_n, C(n,I))$ has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of $S_n$, and we obtain that $\mathrm{Cay}(S_n, C(n,I))$ does not have the Aldous property whenever $n \in I$. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,\{k\}))$ where $2 \le k \le n-2$.

math.CO↗
arXiv2022

Aldous' spectral gap property for normal Cayley graphs on symmetric groups

Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the "normal" case of Aldous' spectral gap conjecture.

math.CO↗
arXiv2022

Regular sets in Cayley graphs

In a graph $Γ$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.

math.CO↗
arXiv2022

Subgroup regular sets in Cayley graphs

Let $Γ$ be a graph with vertex set $V$, and let $a$ and $b$ be nonnegative integers. A subset $C$ of $V$ is called an $(a,b)$-regular set in $Γ$ if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In particular, $(0, 1)$-regular sets and $(1, 1)$-regular sets in $\Ga$ are called perfect codes and total perfect codes in $\Ga$, respectively. A subset $C$ of a group $G$ is said to be an $(a,b)$-regular set of $G$ if there exists a Cayley graph of $G$ which admits $C$ as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group $G$ or any group $G$ of order $4p$ or $pq$ for some primes $p$ and $q$, if a nontrivial subgroup $H$ of $G$ is a $(0, 1)$-regular set of $G$, then it must also be an $(a,b)$-regular set of $G$ for any $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $a$ is even when $|H|$ is odd. A similar result involving $(1, 1)$-regular sets of such groups is also obtained in the paper.

math.CO↗
arXiv2022

Stability of graph pairs involving vertex-transitive graphs

A pair of graphs $(Γ,Σ)$ is said to be stable if the full automorphism group of $Γ\timesΣ$ is isomorphic to the product of the full automorphism groups of $Γ$ and $Σ$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. In this paper, we reduce the study of the stability of any pair of regular graphs $(Γ,Σ)$ with coprime valencies and vertex-transitive $Σ$ to that of $(Γ,K_2)$. Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs $(Γ,Σ)$ with coprime valencies in the case when $Σ$ is vertex-transitve and the stability of $(Γ,K_2)$ is known.

math.CO↗
arXiv2022

Eigenvalues of Cayley graphs

We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.

math.CO↗
arXiv2022

Distance-constrained labellings of Cartesian products of graphs

An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is a mapping $ϕ: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair of vertices $u, v$ of $G$ at distance $i$, we have $|ϕ(u) - ϕ(v)| \geq h_i$. The span of $ϕ$ is the difference between the largest and smallest labels assigned to the vertices of $G$ by $ϕ$, and $λ_{h_1, h_2, \ldots, h_l}(G)$ is defined as the minimum span over all $L(h_1, h_2, \ldots, h_l)$-labellings of $G$. In this paper we study $λ_{h, 1, \ldots, 1}$ for Cartesian products of graphs, where $(h, 1, \ldots, 1)$ is an $l$-tuple with $l \ge 3$. We prove that, under certain natural conditions, the value of this and three related invariants on a graph $H$ which is the Cartesian product of $l$ graphs attain a common lower bound. In particular, the chromatic number of the $l$-th power of $H$ equals this lower bound plus one. We further obtain a sandwhich theorem which extends the result to a family of subgraphs of $H$ which contain a certain subgraph of $H$. All these results apply in particular to the class of Hamming graphs: if $q_1\ge \cdots \ge q_d\ge 2$ and $3\le l\le d$ then the Hamming graph $H=H_{q_1,q_2,\ldots ,q_d}$ satisfies $λ_{q_l,1,\ldots,1}(H) = q_1q_2\ldots q_l-1$ whenever $q_1q_2\ldots q_{l-1}>3(q_{l-1}+1)q_l\ldots q_d$. In particular, this settles a case of the open problem on the chromatic number of powers of the hypercubes.

math.CO↗
arXiv2022

Nowhere-zero 3-flows in Cayley graphs on supersolvable groups

Tutte's 3-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow $2$-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

math.CO↗
arXiv2022

Cubic graphical regular representations of some classical simple groups

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}Ω_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}Ω_{n}^{-}(q)$ ($n=8, 10, 12$), where $q = 2^f$ with $f \ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q \rightarrow \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^{i}-1$ for each $1 \le i < ef$ together with a random involution $y$ gives rise to a cubic GRR, where $e=n-2$ for $\mathrm{P}Ω_{n}^{+}(q)$ and $e=n$ for other groups. Moreover, for sufficiently large $q$, there are elements $x$ satisfying these conditions, and for each of them there exists an involution $y$ such that $\{x,x^{-1},y\}$ produces a cubic GRR. This result together with certain known results in the literature implies that except for $\mathrm{PSL}_2(q)$, $\mathrm{PSL}_3(q)$, $\mathrm{PSU}_3(q)$ and a finite number of other cases, every finite non-abelian simple group contains an element $x$ and an involution $y$ such that $\{x,x^{-1},y\}$ produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR.

math.GR↗
arXiv2021

A Graph Symmetrisation Bound on Channel Information Leakage under Blowfish Privacy

Blowfish privacy is a recent generalisation of differential privacy that enables improved utility while maintaining privacy policies with semantic guarantees, a factor that has driven the popularity of differential privacy in computer science. This paper relates Blowfish privacy to an important measure of privacy loss of information channels from the communications theory community: min-entropy leakage. Symmetry in an input data neighbouring relation is central to known connections between differential privacy and min-entropy leakage. But while differential privacy exhibits strong symmetry, Blowfish neighbouring relations correspond to arbitrary simple graphs owing to the framework's flexible privacy policies. To bound the min-entropy leakage of Blowfish-private mechanisms we organise our analysis over symmetrical partitions corresponding to orbits of graph automorphism groups. A construction meeting our bound with asymptotic equality demonstrates tightness.

cs.IT↗
arXiv2021

On subgroup perfect codes in Cayley graphs

A perfect code in a graph $Γ= (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if there exists a Cayley graph of $G$ which admits $H$ as a perfect code. Equivalently, $H$ is a subgroup perfect code of $G$ if there exists an inverse-closed subset $A$ of $G$ containing the identity element such that $(A, H)$ is a tiling of $G$ in the sense that every element of $G$ can be uniquely expressed as the product of an element of $A$ and an element of $H$. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving $2$-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and $2$-groups.

math.CO↗
arXiv2021

An explicit characterization of arc-transitive circulants

A reductive characterization of arc-transitive circulants was given independently by Kovacs in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arc-transitive circulants and their automorphism groups. Based on this, we give a proof of the fact that arc-transitive circulants are all CI-digraphs.

math.CO↗
arXiv2021

Classification of tetravalent $2$-transitive non-normal Cayley graphs of finite simple groups

A graph $Γ$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(Γ)$ is transitive on the set of vertices of $Γ$ and the set of $s$-arcs of $Γ$, where for an integer $s \ge 1$ an $s$-arc of $Γ$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $Γ$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. $Γ$ is called 2-transitive if it is $(\mathrm{Aut}(Γ), 2)$-arc-transitive but not $(\mathrm{Aut}(Γ), 3)$-arc-transitive. A Cayley graph $Γ$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(Γ)$ and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if $Γ$ is a tetravalent 2-transitive Cayley graph of a finite simple group $G$, then either $Γ$ is normal or $G$ is one of the groups $\mathrm{PSL}_2(11)$, $M_{11}$, $M_{23}$ and $A_{11}$. However, it was unknown whether $Γ$ is normal when $G$ is one of these four groups. In the present paper we answer this question by proving that among these four groups only $M_{11}$ produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly two such graphs which are non-isomorphic and both determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

math.CO↗
arXiv2020

Stability of pair graphs

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(Γ,Σ)$ is stable if $Aut(Γ\timesΣ) \cong Aut(Γ)\times Aut(Σ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. An unstable graph pair $(Γ,Σ)$ is said to be a nontrivially unstable graph pair if $Γ$ and $Σ$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pair of graphs to be stable. We also give a characterization of a pair of graphs $(Γ, Σ)$ to be nontrivially unstable in the case when both graphs are connected and regular with coprime valencies and $Σ$ is vertex-transitive. This characterization is given in terms of the $Σ$-automorphisms of $Γ$, which are a new concept introduced in this paper as a generalization of both automorphisms and two-fold automorphisms of a graph.

math.CO↗
arXiv2020

Canonical double covers of generalized Petersen graphs, and double generalized Petersen graphs

The canonical double cover $\D(Γ)$ of a graph $Γ$ is the direct product of $Γ$ and $K_2$. If $\Aut(\D(Γ))\cong\Aut(Γ)\times\ZZ_2$ then $Γ$ is called stable; otherwise $Γ$ is called unstable. An unstable graph is said to be nontrivially unstable if it is connected, non-bipartite and no two vertices have the same neighborhood. In 2008 Wilson conjectured that, if the generalized Petersen graph $\GP(n,k)$ is nontrivially unstable, then both $n$ and $k$ are even, and either $n/2$ is odd and $k^2\equiv\pm 1 \pmod{n/2}$, or $n=4k$. In this note we prove that this conjecture is true. At the same time we determine all possible isomorphisms among the generalized Petersen graphs, the canonical double covers of the generalized Petersen graphs, and the double generalized Petersen graphs. Based on these we completely determine the full automorphism group of the canonical double cover of $\GP(n,k)$ for any pair of integers $n, k$ with $1 \leqslant k < n/2$.

math.CO↗
arXiv2020

Perfect state transfer in NEPS of complete graphs

Perfect state transfer in graphs is a concept arising from quantum physics and quantum computing. Given a graph $G$ with adjacency matrix $A_G$, the transition matrix of $G$ with respect to $A_G$ is defined as $H_{A_{G}}(t) = \exp(-\mathrm{i}tA_{G})$, $t \in \mathbb{R},\ \mathrm{i}=\sqrt{-1}$. We say that perfect state transfer from vertex $u$ to vertex $v$ occurs in $G$ at time $τ$ if $u \ne v$ and the modulus of the $(u,v)$-entry of $H_{A_G}(τ)$ is equal to $1$. If the moduli of all diagonal entries of $H_{A_G}(τ)$ are equal to $1$ for some $τ$, then $G$ is called periodic with period $τ$. In this paper we give a few sufficient conditions for NEPS of complete graphs to be periodic or exhibit perfect state transfer.

math.CO↗