We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants (girth, diameter), and spectral invariants (the determinant of the Laplace operator, or complexity; bottom non-zero eigenvalue of the Laplace operator). We show ...

Let ρ(G) denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 ≤ g ≤ n. It is proved that ρ(G) ≤ ng (m−n+ 1) and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.

Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y in £$, the vertices $x$ and $y$ are adjacent if and only if $x vee y in {rm Z}(£)$. The basic properties of the ...

CONTEXT An independent association between obesity and preclinical carotid atherosclerosis has been demonstrated, however, the pathophysiological links were not clearly established. Body composition (BC) influences systemic hemodynamics and may participate in the remodeling of common carotid artery (CCA), independently of risk factors. OBJECTIVE This study evaluated the association between CC...

Finite geometry is the study of incidence structures, a system of points and lines with some points on some lines according to an incidence relation, such that the number of points and lines is finite. Generalised quadrangles arose from an effort by Tits [5] to give geometric constructions of the finite simple groups of Lie type. Tits' key observation was that the incidence graph of an n-gon al...

Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G−F has no isolated vertex. The restricted edge connectivity λ′ is theminimumcardinality over all restricted edge cuts.WecallG λ′-optimal if λ′ = ξ , where ξ is theminimum edgedegree inG. Moreover, a λ′-optimal graphG is called ...

We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an Ω̃(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is vali...

Background. Waist girth and BMI are commonly used as markers of cardiometabolic risk. Accumulating data however suggest that sagittal abdominal diameter (SAD) or "abdominal height" may be a better marker of intra-abdominal adiposity and cardiometabolic risk. We aimed to identify cutoffs for SAD using a cardiometabolic risk score. Design. A population-based cross-sectional study. Methods. In 403...

It was conjectured by Caccetta and H aggkvist in 1978 that the girth of every digraph with n vertices and minimum outdegree r is at most dn=re. The conjecture was proved for r = 2 by Caccetta and H aggkvist, for r = 3 by Hamidoune and for r = 4; 5 by Ho ang and Reed. In this paper, the following two main results are proved: 1. The diameter of every strongly connected digraph of order n with g...

We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an Ω̃(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is vali...