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A perfect code in a graph is an independent set of the graph such that every vertex outside the set is adjacent to exactly one vertex in the set. A circulant graph is a Cayley graph of a cyclic group. In this paper we study perfect codes in circulant graphs of degree $p^l - 1$, where $p$ is a prime and $l \ge 1$. We obtain a necessary and sufficient condition for such a circulant graph to admit perfect codes, give a construction of all such circulant graphs which admit perfect codes, and prove a lower bound on the number of distinct perfect codes in such a circulant graph. This extends known results for the case $l=1$ and provides insight on the general problem on the existence and structure of perfect codes in circulant graphs.

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition ${\cal B}$, and let ${\cal D}(\Gamma, {\cal B})$ be the incidence structure with point set ${\cal B}$ and blocks $\{B\} \cup \Gamma_{\cal B}(\alpha)$, for $B \in {\cal B}$ and $\alpha \in B$, where $\Gamma_{\cal B}(\alpha)$ is the set of blocks of ${\cal B}$ containing at least one neighbour of $\alpha$ in $\Gamma$. In this paper we classify all $G$-symmetric graphs $\Gamma$ such that $\Gamma_{\cal B}(\alpha) \ne \Gamma_{\cal B}(\beta)$ for distinct $\alpha, \beta \in B$, the quotient graph of $\Gamma$ with respect to ${\cal B}$ is a complete graph, and ${\cal D}(\Gamma, {\cal B})$ is isomorphic to the complement of a $(G, 2)$-point-transitive linear space., Comment: 20 pages

A graph pair $(\Gamma, \Sigma)$ is called stable if $\aut(\Gamma)\times\aut(\Sigma)$ is isomorphic to $\aut(\Gamma\times\Sigma)$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. A graph is called $R$-thin if distinct vertices have different neighbourhoods. $\Gamma$ and $\Sigma$ are said to be coprime if there is no nontrivial graph $\Delta$ such that $\Gamma \cong \Gamma_1 \times \Delta$ and $\Sigma \cong \Sigma_1 \times \Delta$ for some graphs $\Gamma_1$ and $\Sigma_1$. An unstable graph pair $(\Gamma, \Sigma)$ is called nontrivially unstable if $\Gamma$ and $\Sigma$ are $R$-thin connected coprime graphs and at least one of them is non-bipartite. This paper contributes to the study of the stability of graph pairs with a focus on the case when $\Sigma = C_n$ is a cycle. We give two sufficient conditions for $(\Gamma, C_n)$ to be nontrivially unstable, where $n \ne 4$ and $\Gamma$ is an $R$-thin connected graph. In the case when $\Gamma$ is an $R$-thin connected non-bipartite graph, we obtain the following results: (i) if $(\Gamma, K_2)$ is unstable, then $(\Gamma, C_{n})$ is unstable for every even integer $n \geq 4$; (ii) if an even integer $n \ge 6$ is compatible with $\Gamma$ in some sense, then $(\Gamma, C_{n})$ is nontrivially unstable if and only if $(\Gamma, K_2)$ is unstable; (iii) if there is an even integer $n \ge 6$ compatible with $\Gamma$ such that $(\Gamma, C_{n})$ is nontrivially unstable, then $(\Gamma, C_{m})$ is unstable for all even integers $m \ge 6$. We also prove that if $\Gamma$ is an $R$-thin connected graph and $n \ge 3$ is an odd integer compatible with $\Gamma$, then $(\Gamma, C_{n})$ is stable.

A Cayley graph on the symmetric group $S_n$ is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of $S_n$. For $1\leq r < k < n$, let $C(n,k;r)$ be the set of $k$-cycles of $S_n$ moving every point in $\{1, \ldots, r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989--1005] posed a conjecture which is equivalent to saying that for any $n \ge 5$ and $1\leq r

Let $\Ga = (V, E)$ be a graph and $a, b$ nonnegative integers. An $(a, b)$-regular set in $\Ga$ is a nonempty proper subset $D$ of $V$ such that every vertex in $D$ has exactly $a$ neighbours in $D$ and every vertex in $V \setminus D$ has exactly $b$ neighbours in $D$. A $(0,1)$-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset $D$ of a group $G$ is called an $(a,b)$-regular set of $G$ if it is an $(a, b)$-regular set in some Cayley graph of $G$, and an $(a, b)$-regular set in a Cayley graph of $G$ is called a subgroup $(a, b)$-regular set if it is also a subgroup of $G$. In this paper we study $(a, b)$-regular sets in Cayley graphs with a focus on $(0, k)$-regular sets, where $k \ge 1$ is an integer. Among other things we determine when a non-trivial proper normal subgroup of a group is a $(0, k)$-regular set of the group. We also determine all subgroup $(0, k)$-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of $n$ copies of the cycle of length $p$ to admit $(0, k)$-regular sets, where $p$ is an odd prime. Our results generalize several known results from perfect codes to $(0, k)$-regular sets., Comment: 27 pages

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 $2 \le k \le n-2$., Comment: 27 pages

A pair of graphs $(\Gamma,\Sigma)$ is said to be stable if the full automorphism group of $\Gamma\times\Sigma$ is isomorphic to the product of the full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. In this paper, we reduce the study of the stability of any pair of regular graphs $(\Gamma,\Sigma)$ with coprime valencies and vertex-transitive $\Sigma$ to that of $(\Gamma,K_2)$. Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs $(\Gamma,\Sigma)$ with coprime valencies in the case when $\Sigma$ is vertex-transitve and the stability of $(\Gamma,K_2)$ is known., Comment: 9 pages

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest on this question dates back to early 1980s, when P.~J.~Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by L.~Babai in 1985. Then Babai posed the open problem of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problem by constructing an infinite family of $s$-arc-transitive digraphs for each integer $s\geq2$, and an infinite family of vertex-primitive digraphs, respectively, both of whose adjacency matrices are non-diagonalizable.

We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow., Comment: 15 pages

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., Comment: Final version published in European Journal of Combinatorics

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.

A radio labelling of a graph $G$ is a mapping $f : V(G) \rightarrow \{0, 1, 2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number $rn(G)$ of $G$ is the smallest integer $k$ such that $G$ admits a radio labelling $f$ with $\max\{f(v):v \in V(G)\} = k$. The weight of a tree $T$ from a vertex $v \in V(T)$ is the sum of the distances in $T$ from $v$ to all other vertices, and a vertex of $T$ achieving the minimum weight is called a weight center of $T$. It is known that any tree has one or two weight centers. A tree is called a two-branch tree if the removal of all its weight centers results in a forest with exactly two components. In this paper we obtain a sharp lower bound for the radio number of two-branch trees which improves a known lower bound for general trees. We also give a necessary and sufficient condition for this improved lower bound to be achieved. Using these results, we determine the radio number of two families of level-wise regular two-branch trees., Comment: 29 pages, 3 figures

We correct the statements of two theorems and two corollaries in our paper [On subgroup perfect codes in Cayley graphs, European J. Combin. 91 (2021) 103228]. Proofs of these theorems and three other results are given as well., Comment: This is the final version published in European J. Combin. 101 (2022) 103461. arXiv admin note: text overlap with arXiv:2006.11104

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 $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}\Omega_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}\Omega_{n}^{-}(q)$ ($n=8, 10, 12$), 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 $\mathrm{P}\Omega_{n}^{+}(q)$ 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.

Let $\Gamma$ 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 $\Gamma$ 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., Comment: 9 pages

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 $(\Gamma,\Sigma)$ is stable if $Aut(\Gamma\times\Sigma) \cong Aut(\Gamma)\times Aut(\Sigma)$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. An unstable graph pair $(\Gamma,\Sigma)$ is said to be a nontrivially unstable graph pair if $\Gamma$ and $\Sigma$ 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 $(\Gamma, \Sigma)$ to be nontrivially unstable in the case when both graphs are connected and regular with coprime valencies and $\Sigma$ is vertex-transitive. This characterization is given in terms of the $\Sigma$-automorphisms of $\Gamma$, which are a new concept introduced in this paper as a generalization of both automorphisms and two-fold automorphisms of a graph., Comment: 22 pages

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 $\tau$ if $u \ne v$ and the modulus of the $(u,v)$-entry of $H_{A_G}(\tau)$ is equal to $1$. If the moduli of all diagonal entries of $H_{A_G}(\tau)$ are equal to $1$ for some $\tau$, then $G$ is called periodic with period $\tau$. In this paper we give a few sufficient conditions for NEPS of complete graphs to be periodic or exhibit perfect state transfer., Comment: This is the final version that will appear in Discrete Applied Mathematics

Given graphs $G$ and $H$, the generalized Tur\'{a}n 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 ${\rm S}_n$., Comment: 16 pages, 1 figure

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. In this paper we describe the structure of maximal non-trivial $t$-intersecting families of $k$-dimensional subspaces of $V$ with large size. We also determine the non-trivial $t$-intersecting families with maximum size. In the special case when $t=1$ our result gives rise to the well-known Hilton-Milner Theorem for vector spaces., Comment: 26 pages

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., Comment: 11 pages, 5 figures; accepted to IEEE Transactions on Information Theory

A perfect code in a graph $\Gamma = (V, E)$ 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., Comment: This is the corrected version of our paper (arXiv:2006.11104v1) under the same title published in European Journal of Combinatorics 91 (2021) 103228 (https://doi.org/10.1016/j.ejc.2020.103228)

An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is a mapping $\phi: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair of vertices $u, v$ of $G$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq h_i$. The span of $\phi$ is the difference between the largest and smallest labels assigned to the vertices of $G$ by $\phi$, and $\lambda_{h_1, h_2, \ldots, h_l}(G)$ is defined as the minimum span over all $L(h_1, h_2, \ldots, h_l)$-labellings of $G$. In this paper we study $\lambda_{h, 1, \ldots, 1}$ 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 $\lambda_{q_l,1,\ldots,1}(H) = q_1q_2\ldots q_l-1$ 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., Comment: Final version published in Discrete Applied Mathematics

In a graph $\Gamma$ 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., Comment: 12 pages

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such that for blocks $B, C \in {\cal B}$ adjacent in the quotient graph $\Gamma_{{\cal B}}$ of $\Gamma$ relative to ${\cal B}$, exactly one vertex of $B$ has no neighbour in $C$, then $\Gamma$ is called an almost multicover of $\Gamma_{{\cal B}}$. In this case an incidence structure with point set ${\cal B}$ arises naturally, and it is a $(G, 2)$-point-transitive and $G$-block-transitive 2-design if in addition $\Gamma_{{\cal B}}$ is a complete graph. In this paper we classify all $G$-symmetric graphs $\Gamma$ such that (i) ${\cal B}$ has block size $|B| \ge 3$; (ii) $\Gamma_{{\cal B}}$ is complete and almost multi-covered by $\Gamma$; (iii) the incidence structure involved is a linear space; and (iv) $G$ contains a regular normal subgroup which is elementary abelian. This classification together with earlier results in [A. Gardiner and C. E. Praeger, Australas. J. Combin. 71 (2018) 403--426], [M.~Giulietti et al., J. Algebraic Combin. 38 (2013) 745--765] and [T. Fang et al., Electronic J. Combin. 23 (2) (2016) P2.27] completes the classification of symmetric graphs satisfying (i) and (ii)., Comment: This is the final version

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.

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if there exists a Cayley graph of $G$ which admits $C$ as a perfect code. A group $G$ is said to be code-perfect if every proper subgroup of $G$ is a perfect code of $G$. In this paper we prove that a group is code-perfect if and only if it has no elements of order $4$. We also prove that a proper subgroup $H$ of an abelian group $G$ is a perfect code of $G$ if and only if the Sylow $2$-subgroup of $H$ is a perfect code of the Sylow $2$-subgroup of $G$. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian $2$-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group., Comment: This is the final version to be published in SIAM Journal on Discrete Mathematics, 18 pp

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., Comment: The final version

The canonical double cover $\D(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\Aut(\D(\Gamma))\cong\Aut(\Gamma)\times\ZZ_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ 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 $1 \leqslant k < n/2$., Comment: This is the final version that will appear in Journal of Graph Theory

The canonical double cover $\mathrm{D}(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\mathrm{Aut}(\mathrm{D}(\Gamma))=\mathrm{Aut}(\Gamma)\times\mathbb{Z}_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. In this paper we prove that every circulant graph of odd prime order is stable and there is no arc-transitive nontrivially unstable circulant graph. The latter answers a question of Wilson in 2008. We also give infinitely many counterexamples to a conjecture of Maru\v{s}i\v{c}, Scapellato and Zagaglia Salvi in 1989 by constructing a family of stable circulant graphs with compatible adjacency matrices.

We derive upper and lower bounds for the vertex-isoperimetric number of the incidence graphs of unitals and determine its order of magnitude. In the case when a unital contains sufficiently large arcs, these bounds agree and give rise to the precise value of this parameter. In particular, we obtain the exact value of the vertex-isoperimetric number of the incidence graphs of classical unitals and a certain subfamily of BM-unitals. In the case when the maximum size of arcs in the unital is relatively small, we obtain an upper bound for this parameter in terms of the vertex-isoperimetric number of the incidence graph. We also determine the exact value of the vertex-isoperimetric number of the non-incidence graph of any unital.

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such that every vertex of $V$ is adjacent to exactly one vertex in $C$. A perfect code in the Hamming graph $H(n, q)$ agrees with a $q$-ary perfect 1-code of length $n$ in the classical setting. In this paper we give a necessary and sufficient condition for a circulant graph of degree $p-1$ to admit a perfect code, where $p$ is an odd prime. We also obtain a necessary and sufficient condition for a circulant graph of order $n$ and degree $p^l-1$ to have a perfect code, where $p$ is a prime and $p^l$ the largest power of $p$ dividing $n$. Similar results for total perfect codes are also obtained in the paper.

Let $\Gamma_{n,q}$ be the point-hyperplane incidence graph of the projective space $\operatorname{PG}(n,q)$, where $n \ge 2$ is an integer and $q$ a prime power. We determine the order of magnitude of $1-i_V(\Gamma_{n,q})$, where $i_V(\Gamma_{n,q})$ is the vertex-isoperimetric number of $\Gamma_{n,q}$. We also obtain the exact values of $i_V(\Gamma_{2,q})$ and the related incidence-free number of $\Gamma_{2,q}$ for $q \le 16$.

A graph $\Gamma$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(\Gamma)$ is transitive on the set of vertices of $\Gamma$ and the set of $s$-arcs of $\Gamma$, where for an integer $s \ge 1$ an $s$-arc of $\Gamma$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $\Gamma$ 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 $1 \le i \le s-1$. $\Gamma$ is called 2-transitive if it is $(\mathrm{Aut}(\Gamma), 2)$-arc-transitive but not $(\mathrm{Aut}(\Gamma), 3)$-arc-transitive. A Cayley graph $\Gamma$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(\Gamma)$ and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if $\Gamma$ is a tetravalent 2-transitive Cayley graph of a finite simple group $G$, then either $\Gamma$ 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 $\Gamma$ 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., Comment: Final version published in Bulletin of the Australian Mathematical Society

A retract of a graph $\Gamma$ is an induced subgraph $\Psi$ of $\Gamma$ such that there exists a homomorphism from $\Gamma$ to $\Psi$ whose restriction to $\Psi$ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph $\Gamma$ is $G$-symmetric if $G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of $\Gamma$ admits a nontrivial partition that is preserved by $G$, then $\Gamma$ is an imprimitive $G$-symmetric graph. In this paper cores of imprimitive symmetric graphs $\Gamma$ of order a product of two distinct primes are studied. In many cases the core of $\Gamma$ is determined completely. In other cases it is proved that either $\Gamma$ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.

Metacirculants are a basic and well-studied family of vertex-transitive graphs, and weak metacirculants are generalizations of them. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. This paper is devoted to the study of weak metacirculants with odd prime power order. We first prove that a weak metacirculant of odd prime power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. We then prove that for any odd prime $p$ and integer $\ell\geq 4$, there exist weak metacirculants of order $p^\ell$ which are Cayley graphs but not Cayley graphs of any metacyclic group; this answers a question in Li et al. (2013). We construct such graphs explicitly by introducing a construction which is a generalization of generalized Petersen graphs. Finally, we determine all smallest possible metacirculants of odd prime power order which are Cayley graphs but not Cayley graphs of any metacyclic group.

Given a graph $\Gamma$, a subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$, and a subset $C$ of $V(\Gamma)$ is called a total perfect code in $\Gamma$ if every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions., Comment: This is the final version that will appear in SIAM J. Discrete Math

A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that $|f(u)-f(v)|\geq d + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$, where $d$ is the diameter of $G$ and $d(u,v)$ the distance between $u$ and $v$ in $G$. The radio number of $G$ is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees., Comment: 19 pages, 7 figures. This is the final version accepted in Discrete Applied Mathematics