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Let G be a finite group, let
${\text{Irr}}(G)$ be the set of all irreducible complex characters of G and let
$\chi \in {\text{Irr}}(G)$. Define the codegrees,
${\text{cod}}(\chi ) = |G: {\text{ker}}\chi |/\chi (1)$ and
${\text{cod}}(G) = \{{\text{cod}}(\chi ) \mid \chi \in {\text{Irr}}(G)\} $. We show that the simple group
${\text{PSL}}(2,q)$, for a prime power
$q>3$, is uniquely determined by the set of its codegrees.
Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.
A graph $\Gamma $ is called $(G, s)$-arc-transitive if $G \le \text{Aut} (\Gamma )$ 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 $1 \le i \le s-1$. A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$-arc-transitive but not $(\text{Aut} (\Gamma ), 3)$-arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved 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 $\text{PSL}_2(11)$, $\text{M} _{11}$, $\text{M} _{23}$ and $A_{11}$. However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are 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.
For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.
For a finite group G, let
$\Delta (G)$
denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph
$\Gamma $
in which the chromatic number of every induced subgraph
$\Delta $
of
$\Gamma $
equals the clique number of
$\Delta $
. We show that the character graph
$\Delta (G)$
of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of
$\Delta (G)$
is at most three.
We consider the graph
$\Gamma _{\text {virt}}(G)$
whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph
$\Delta _{\text {virt}}(G)$
obtained from
$\Gamma _{\text {virt}}(G)$
by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that
$\Delta _{\operatorname {\mathrm {virt}}}(G)$
has precisely t connected components. Moreover, we study the graph
$\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$
, whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph
$\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$
obtained removing the isolated vertices is connected and has diameter at most 3.
We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.
We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning
$ O(\sqrt t )$
vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown–Erdős–Sós conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi’s theorem and the density Hales–Jewett theorem.
This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].
For a finite group
$G$
, let
$\unicode[STIX]{x1D6E5}(G)$
denote the character graph built on the set of degrees of the irreducible complex characters of
$G$
. In this paper, we obtain a necessary and sufficient condition which guarantees that the complement of the character graph
$\unicode[STIX]{x1D6E5}(G)$
of a finite group
$G$
is a nonbipartite Hamiltonian graph.
With entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.
We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].
An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.
This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.
Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of $G$. The character degree graph $\unicode[STIX]{x1D6E5}(G)$ associated to $G$ is a graph whose vertex set is $\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides $\unicode[STIX]{x1D712}(1)$ for some $\unicode[STIX]{x1D712}\in \text{Irr}(G)$. We prove that $\unicode[STIX]{x1D6E5}(G)$ is $k$-regular for some natural number $k$ if and only if $\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.
We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$-minimal expansions of $\mathbb{R}$, and show that it does not hold in $\mathbb{R}_{\exp }$. This provides a new combinatorial characterization of polynomial boundedness for $o$-minimal structures. We also prove an analog for relations definable in $P$-minimal structures, in particular for the field of the $p$-adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society366(9) (2014), 5043–5065), we show that in distal structures the upper bound for $k$-ary definable relations is given by the exponential tower of height $k-1$.
For a group $G$, let $\unicode[STIX]{x1D6E4}(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Let $\unicode[STIX]{x1D6E4}^{\ast }(G)$ be the subgraph of $\unicode[STIX]{x1D6E4}(G)$ that is induced by all the vertices of $\unicode[STIX]{x1D6E4}(G)$ that are not isolated. We prove that if $G$ is a 2-generated noncyclic abelian group, then $\unicode[STIX]{x1D6E4}^{\ast }(G)$ is connected. Moreover, $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=2$ if the torsion subgroup of $G$ is nontrivial and $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=\infty$ otherwise. If $F$ is the free group of rank 2, then $\unicode[STIX]{x1D6E4}^{\ast }(F)$ is connected and we deduce from $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$ that $\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(F))=\infty$.
In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspace, each network admissible system restricted to that subspace is a dynamical system consistent with a smaller network called a quotient network. We characterize networks such that: the network is a subnetwork of its fundamental network, and the network is a fundamental network. Moreover, we prove that the fundamental network construction preserves the quotient relation and it transforms the subnetwork relation into the quotient relation. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.
For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).
Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).
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, for instance, that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (nonempty) subsets summing to $0$, and that a random colouring of the edges of a complete graph contains a monochromatic triangle.