Vertex-primitive s-arc-transitive digraphs of linear groups

We study G-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups G with socle ${\mathrm{PSL}}_n(q)$. We prove that $s?2$ for such digraphs, which provides the first step in determining an upper bound on s for all the vertex-primitive s-arc-transitive digraphs.     

Graphs with nullity 2c(G)+p(G)−1

Let G be a connected graph with $n(G)$ vertices and $e(G)$ edges. The nullity of G, denoted by $\eta (G)$, is the multiplicity of eigenvalue zero of the adjacency matrix of G. Ma, Wong and Tian (2016) proved that $\eta (G)\le 2c(G)+p(G)?1$ unless G is a cycle of order a multiple of 4, where $c(G)=e(G)?n(G)+1$ is the elementary cyclic number of G and $p(G)$ is the number of leaves of G. Recently, Chang, Chang and Zheng (2020) characterized the leaf-free graphs with nullity $2c(G)?1$, thus leaving the problem to characterize connected graphs G with nullity $2c(G)+p(G)?1$ when $p(G)\ne 0$. In this paper, we solve this problem completely.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Spanning trees and spanning closed walks with small degrees

Let G be a graph and let f be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S?V(G)$, $\omega (G?S)<{\sum }_{v\in S}(f(v)?2)+2+\omega (G[S])$, then G has a spanning tree T containing an arbitrary given matching such that for each vertex v, $d_T(v)\le f(v)$, where $\omega (G?S)$ denotes the number of components of $G?S$ and $\omega (G[S])$ denotes the number of components of the induced subgraph $G[S]$ with the vertex set S. This is an improvement of several results. Next, we prove that if for all $S?V(G)$, $\omega (G?S)\le {\sum }_{v\in S}(f(v)?1)+1$, then G admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex v at most $f(v)$ times. This result solves a long-standing conjecture due to Jackson and Wormald (1990).     

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.