An edge cut of a connected graph is 5-restricted if its removal leaves every component having order at least five. Graphs that contain 5-restricted edge cuts are characterized in this paper. As a result, it is shown that a connected graph G of order at least 13 contains 5-restricted edge cuts if and only if G \ v contains a component of order at least five for every vertex v of graph G.

For a permutation f of the vertex set V (G) of a connected graph G, let δf (x, y)= |d(x, y)− d(f (x), f (y))|. Define the displacement δf (G) of G with respect to f to be the sum of δf (x, y) over all unordered pairs {x, y} of distinct vertices of G. Let π(G) denote the smallest positive value of δf (G) among the n! permutations f of V (G). In this note, we determine all trees T with π(T )= 2 o...

We pursue the hypothesis that neuronal placement in animals minimizes wiring costs for given functional constraints, as specified by synaptic connectivity. Using a newly compiled version of the Caenorhabditis elegans wiring diagram, we solve for the optimal layout of 279 nonpharyngeal neurons. In the optimal layout, most neurons are located close to their actual positions, suggesting that wirin...

Let T ∆ n denote the set of trees of order n, in which the degree of each vertex is bounded by some integer ∆. Suppose that every tree in T ∆ n is equally likely. We show that the number of vertices of degree j in T ∆ n is asymptotically normal with mean (μj + o(1))n and variance (σj + o(1))n, where μj , σj are some constants. As a consequence, we give estimate to the value of the general Zagre...

The general Randić index Rα(G) is the sum of the weight d(u)d(v)α over all edges uv of a graph G, where α is a real number and d(u) is the degree of the vertex u of G. In this paper, for any real number α = 0, the first three minimum general Randić indices among trees are determined, and the corresponding extremal trees are characterized.

Let Γ be an infinite, locally finite, connected graph with distance function δ. Given a ray P in Γ and a constant C ≥ 1, a vertex-sequence {xn}n=0 ⊆ V P is said to be regulated by C if, for all n ∈ N, xn+1 never precedes xn on P , each vertex of P appears at most C times in the sequence, and δP (xn, xn+1) ≤ C.

Sensory deprivation during a critical period reduces spine motility and disrupts receptive field structure of layer 2/3 neurons in rat barrel cortex. To determine the locus of plasticity, we used laser scanning photostimulation, allowing us to rapidly map intracortical synaptic connectivity in brain slices. Layer 2/3 neurons differed in their spatial distributions of presynaptic partners: neuro...

All the polygonalisation algorithms we can find in the literature proceed on 4-connected or 8-connected discrete arcs. In this article, we aim to polygonalise “thick arcs”. A first step consists in giving a definition of such arcs based on morphological properties. In a second step, we propose two methods in order to polygonalise such arcs. The first one is based on a squelettisation of the arc...

Let G be a connected graph of order n, a and b be integers such that 1 ≤ a ≤ b and 2 ≤ b, and f : V (G) → {a, a + 1, . . . , b} be a function such that ∑ (f(x);x ∈ V (G)) ≡ 0(mod2). We prove the following two results: (i) If the binding number of G is greater than (a+b−1)(n−1)/(an−(a+b)+3) and n ≥ (a+b)2/a, then G has an f factor; (ii) if the minimum degree of G is greater than (bn−2)/(a+b), an...

A connected graph of order n is bicyclic if it has n+1 edges. He et al. [C.X. He, J.Y. Shao, J.L. He, On the Laplacian spectral radii of bicyclic graphs, Discrete Math. 308 (2008) 5981–5995] determined, among the n-vertex bicyclic graphs, the first four largest Laplacian spectral radii together with the corresponding graphs (six in total). It turns that all these graphs have the spectral radius...