Let G be an r-regular graph of order n and independence number α(G). We show that if G has odd girth 2k + 3 then α(G) ≥ n1−1/kr1/k . We also prove similar results for graphs which are not regular. Using these results we improve on the lower bound of Monien and Speckenmeyer, for the independence number of a graph of order n and odd girth 2k + 3. AMS Subject Classification. 05C15 §

A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.

Group equivariant and steerable convolutional neural networks (regular and steerable G-CNNs) have recently emerged as a very effective model class for learning from signal data such as 2D and 3D images, video, and other data where symmetries are present. In geometrical terms, regular G-CNNs represent data in terms of scalar fields (“feature channels”), whereas the steerable G-CNN can also use v...

The symmetric algebra S(g) over a Lie algebra g has the structure of a Poisson algebra. Assume g is complex semisimple. Then results of Fomenko–Mischenko (translation of invariants) and A. Tarasev construct a polynomial subalgebra H = C[q1, . . . , qb] of S(g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of g. Let G be the adjoint group of g and let l = ...

Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d− 1)2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresp...

Let K denote the class of functions g(z) = z + a 2 z 2 + ··· which are regular and univalently convex in the unit disc E. In the present note, we prove that if f is regular in E, f (0) = 0, then for g ∈ K, f (z) + αzf (z) ≺ g(z) + αzg (z) in E implies that f (z) ≺ g(z) in E, where α > 0 is a real number and the symbol " ≺ " stands for subordination.

Let ρ : G GL(n, IF) be a representation of a finite group over the field IF of characteristic p, and h1 , . . . , hm ∈IF[V ]G ivariant polynomials that form a regular sequence in IF[V ]. In this note we introduce a tool to study the problem of whether they form a regular sequence in IF[V ]G. Examples show they need not. We define the cohomology of G with coefficients in the Koszul complex

The cyclic matching sequenceability of a simple graph G, denoted cms ( G ) , is the largest integer s for which there exists ordering edges so that every set consecutive forms matching. In this paper we consider minimum k-regular graphs. We completely determine 2-regular graphs, and give bounds k ⩾ 3.

A regular map is a symmetric embedding of a graph (or multigraph) on some closed surface. In this paper we consider the genus spectrum for such maps on orientable surfaces, with simple underlying graph. It is known that for some positive integers g, there is no orientably-regular map of genus g for which both the map and its dual have simple underlying graph, and also that for some g, there is ...

A theorem of A. Schrijver asserts that a d–regular bipartite graph on 2n vertices has at least ( (d− 1)d−1 dd−2 )n perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive in...