In this paper we shall present a natural generalisation of the notion of automorphism of a graph or digraph G, namely a two-fold automorphism. This is a pair (α, β) of permutations of the vertex set V(G) which acts on ordered pairs of vertices of G in the natural way. The action of (α, β) on all such ordered pairs gives a graph which is two-fold isomorphic to G. When α = β the two-fold automorp...

Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that AdG Γ has the same Zariski closure as AdG. If α : Γ → GLn(R) is any finite-dimensional representation of Γ, we show that α virtually extends to a continuous representation σ of G. Furthermore, the image of σ is contained in the Zariski closure of the image of α. When Γ is not discrete, the same conclusions are t...

The well-known Chvátal–Erdős Theorem states that every graph G of order at least three with α(G) ≤ κ(G) has a hamiltonian cycle, where α(G) and κ(G) are the independence number and the connectivity of G, respectively. Oberly and Sumner [J. Graph Theory 3 (1979), 351–356] have proved that every connected, locally-connected claw-free graph of order at least three has a hamiltonian cycle. We study...

Let G be a graph on n vertices with independence number α. What is the largest k-connected subgraph that G must contain? We prove that if n is sufficiently large (n ≥ αk + 1 will do), then G contains a k-connected subgraph on at least n/α vertices. This is sharp, since G might be the disjoint union of α equally-sized cliques. For k ≥ 3 and α = 2, 3, we shall prove that the same result holds for...

A lower bound for the independence number of a graph is a graph invariant l such that, for every graph G, l(G) ≤ α(G). Similarly, an upper bound for the independence number is a graph invariant u such that, for every graph G, α(G) ≤ u(G). Many efficiently computable upper and lower bounds, called α-bounds here, have been published and these are surveyed in the following section. They can be use...

Drosophila photoreceptor cells use the ubiquitous G-protein-mediated phospholipase C (PLC) cascade to achieve ultimate single-photon sensitivity. This is manifested in the single-photon responses (quantum bumps). In photoreceptor cells, dark activation of G(q)α molecules occurs spontaneously and produces unitary dark events (dark bumps). A high rate of spontaneous G(q)α activation and dark bump...

For At(x) = f(x) − t g(x), we consider the set { ∑ At(α)=0 h(α) : t ∈ Q}. The polynomials f(x), g(x) are in Z[x], with only mild restrictions, and h(α) is the Weil height of α. We show that this set is dense in [d,∞) for some effectively computable limit point d.

The paper considers the eigenvalue problem −∆u − αu + λg(x)u = 0 with u ∈ H(R ), u = 0, where α, λ ∈ R and g(x) ≡ 0 on Ω, g(x) ∈ (0, 1] on R \ Ω and lim |x |→+∞ g(x) = 1 for some bounded open set Ω ∈ RN . Given α > 0, does there exist a value of λ > 0 for which the problem has a positive solution? It is shown that this occurs if and only if α lies in a certain interval (Γ, ξ1) and that in this ...

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α ∈ [0, 1], write Aα (G) for the matrix Aα (G) = αD (G) + (1− α)A (G) . Let α0 (G) be the smallest α for which Aα(G) is positive semidefinite. It is known that α0 (G) ≤ 1/2. The main results of this paper are: (1) if G is d-regular then α0 = −λmin(A(G)) d− λmin(A(G)) , where λmin(...

We study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph. Our work is based on, and refines de Klerk and Pasechnik’s approach to approximating the stability number via copositive programming (SIAM J. Optim. 12 (2002), 875–892). We provide a closed-form expression for the values computed by the linear programming approximati...