Angeles Carmona

On the order and size of s -geodetic digraphs with given connectivity

Abstract A digraph G = (V, E) with diameter D is said to be s-geodetic, for 1 ⩽ s ⩽ D, if between any pair of (not necessarily different) vertices x, y ϵ V there is at most one x → y path of length ⩽ s. Thus, any loopless digraph is at least 1-geodetic. A similar definition applies for a graph G, but in this case the concept is closely related to its girth g, for then G is s-geodetic with s = ⌊(g − 1)/2⌋. The case s = D corresponds to the so-called (strongly) geodetic (di)graphs. Some recent results have shown that if the order n of a (di)graph is big enough, then its vertex connectivity attains its maximum value. In other words, the (di)graph is maximally connected. Moreover, a similar result involving the size m (number of edges) and edge-connectivity applies. In this work we mainly show that the same conclusions can be reached if the order or size of a s-geodetic (di)graph is small enough. As a corollary, we find some Chartrand-type conditions to assure maximum connectivities. For example, when s ⩾ 2, a s-geodetic digraph is maximally connected if δ ⩾ ⌈ 5 n/2−1 ⌉ . Under similar hypotheses it is also shown that stronger measures of connectivity, such as the so-called super-connectivity, attain also their maximum possible values.

Estimation of Fekete points

We aim here at presenting a new procedure to numerically estimate the Fekete points of a wide variety of compact sets in R^3. We understand the Fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of N particles. The compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves. In order to determine a good initial configuration to start the search of the Fekete points of these objects, we construct a sequence of approximating regular surfaces. Our algorithm is based on the concept of disequilibrium degree, which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration. Moreover, the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy. The numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than N^3.

Extraconnectivity of graphs with large minimum degree and girth

Abstract The extraconnectivity κ ( n ) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to κ (0) and κ (1), respectively. This paper gives sufficient conditions, relating the diameter D , the girth g , and the minimum degree δ of a graph, to assure maximum extraconnectivity. For instance, if D ⩽ g - n + 2( δ - 3), for n ⩾ 2 δ + 4 and g ⩾ n + 5, then the value of κ ( n ) is ( n - 1) δ - 2 n , which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity λ ( n ), is also discussed.

Potential Theory for Schrödinger operators on finite networks

We aim here at analyzing the fundamental properties of positive semidefinite Schrodinger operators on networks. We show that such operators correspond to perturbations of the combinatorial Laplacian through 0-order terms that can be totally negative on a proper subset of the network. In addition, we prove that these discrete operators have analogous properties to the ones of elliptic second order operators on Riemannian manifolds, namely the monotonicity, the minimum principle, the variational treatment of Dirichlet problems and the condenser principle. Unlike the continuous case, a discrete Schrodinger operator can be interpreted as an integral operator and therefore a discrete Potential Theory with respect to its associated kernel can be built. We prove that the Schrodinger kernel satisfies enough principles to assure the existence of equilibrium measures for any proper subset. These measures are used to obtain systematic expressions of the Green and Poisson kernels associated with Dirichlet problems.

On the superconnectivity and the conditional diameter of graphs and digraphs

Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D ≤ 2l- 1, then G has maximum connectivity (κ = δ), and if D ≤ 2l, then it attains maximum edge-connectivity (λ = δ), where l is a parameter which can be thought of as a generalization of the girth of a graph. In this paper, we study some similar conditions for a digraph to attain high connectivities, which are given in terms of what we call the conditional diameter or P-diameter of G. This parameter measures how far apart can be a pair of subdigraphs satisfying a given property P, and, hence, it generalizes the standard concept of diameter. As a corollary, some new sufficient conditions to attain maximum connectivity or edge-connectivity are derived. It is also shown that these conditions can be slightly relaxed when the digraphs are bipartite. The case of (undirected) graphs is managed as a corollary of the above results for digraphs. In particular, since l ≥ 1, some known results of Plesnik and Znam are either reobtained or improved. For instance, it is shown that any graph whose line graph has diameter D = 2 (respectively, D ≤ 3) has maximum connectivity (respectively, edge-connectivity.) Moreover, for graphs with even girth and minimum degree large enough, we obtain a lower bound on their connectivities.

Superconnectivity of bipartite digraphs and graphs

Abstract A maximally connected digraph G is said to be super-κ if all its minimum disconnecting sets are trivial. Analogously, G is called super-λ if it is maximally arc-connected and all its minimum arc-disconnecting sets are trivial. It is first proved that any bipartite digraph G with diameter D is super-κ if D ⩽ 2l − 1, and it is super-λ if D ⩽ 2l, where l denotes a parameter related to the number of short paths. These results allow us to show that if the order of a bipartite digraph G is big enough then superconnectivity is attained. For instance, if G is d-regular and has diameter D = 3 and l ⩾ 1, then G is super-λ if n > 4d; and if D = 4 and l ⩾ 2, then G is super-κ if n > 4d2. In these cases the results are proved to be best possible. Similar results are given for bipartite (undirected) graphs. (For a graph it turns out that l = (g − 2)/2, where g stands for the girth.)

Computational cost of the Fekete problem I: The Forces Method on the 2-sphere

Here, we study the computational complexity of the Fekete point problem. Namely, we give an exhaustive description of the main properties of an algorithm for the minimization of the logarithmic potential energy on the 2-sphere, and we characterize the probability distribution of the cost of the different minima. In particular, we show that a local minimum can be found with an average cost of about O(N^2^.^8).

Distance connectivity in graphs and digraphs

Let G = (V, A) be a digraph with diameter D 6= 1. For a given integer 2 ≤ t ≤ D, the t-distance connectivity κ(t) of G is the minimum cardinality of an x → y separating set over all the pairs of vertices x, y which are at distance d(x, y) ≥ t. The t-distance edge connectivity λ(t) of G is defined similarly. The t-degree of G, δ(t), is the minimum among the out-degrees and in-degrees of all vertices with (outor in-)eccentricity at least t. A digraph is said to be maximally distance connected if κ(t) = δ(t) for all values of t. In this paper we give a construction of a digraph having D − 1 positive arbitrary integers c2 ≤ . . . ≤ cD, D > 3, as the values of its t-distance connectivities κ(2) = c2, . . . , κ(D) = cD. Besides, a digraph that shows the independence of the parameters κ(t), λ(t), and δ(t) is constructed. Also we derive some results on the distance connectivities of digraphs, as well as sufficient conditions for a digraph to be maximally distance connected. Similar results for (undirected) graphs are presented.

Shortest Paths in Distance-regular Graphs

We aim here to introduce a new point of view of the Laplacian of a graph, ?. With this purpose in mind, we consider L as a kernel on the finite space V(?), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(?). If ? is a proper set of a suitable host graph, then the equilibrium problem for ? can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of ?. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs.

Solving Dirichlet and Poisson problems on graphs by means of equilibrium measures

We aim here at obtaining an explicit expression of the solution of the Dirichlet and Poisson problems on graphs. To this end, we consider the Laplacian of a graph as a kernel on the vertex set, V , in the framework of Potential Theory. Then, the properties of such a kernel allow us to obtain for each proper vertex subset the equilibrium measure that solves the so-called equilibrium problem . As a consequence, the Green function of the Dirichlet problems, the generalized Green function of the Poisson problems and the solution of the condenser principle are obtained solely in terms of equilibrium measures for suitable subsets. In particular, we get a formula for the effective resistance between any pair of vertices of a graph. Specifically, r xy = 1/n (v x (y) + v y (x)) , where v z denotes the equilibrium measure for the set V - {z} . In any case, the equilibrium measure for a proper subset is accomplished by solving a Linear Programming Problem.