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Social Network Analysis

Distance-Regular Graphs

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

Uniquely identifying the edges of a graph: The edge metric dimension

Given a set of vertices S={v"1,v"2,...,v"k} of a connected graph G, the metric representation of a vertex v of G with respect to S is the vector r(v|S)=(d(v,v"1),d(v,v"2),...,d(v,v"k)), where d(v,v"i), i@?{1,...,k} denotes the distance between v and v"i. S is a resolving set for G if for every pair of distinct vertices u,v of G, r(u|S) r(v|S). The metric dimension of G, dim(G), is the minimum cardinality of any resolving set for G. Let G and H be two graphs of order n"1 and n"2, respectively. The corona product G@?H is defined as the graph obtained from G and H by taking one copy of G and n"1 copies of H and joining by an edge each vertex from the ith-copy of H with the ith-vertex of G. For any integer k>=2, we define the graph G@?^kH recursively from G@?H as G@?^kH=(G@?^k^-^1H)@?H. We give several results on the metric dimension of G@?^kH. For instance, we show that given two connected graphs G and H of order n"1>=2 and n"2>=2, respectively, if the diameter of H is at most two, then dim(G@?^kH)=n"1(n"2+1)^k^-^1dim(H). Moreover, if n"2>=7 and the diameter of H is greater than five or H is a cycle graph, then dim(G@?^kH)=n"1(n"2+1)^k^-^1dim(K"1@?H).

On the metric dimension of corona product graphs

On the strong metric dimension of corona product graphs and join graphs

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension.

Ismael G. Yero

Papers

Uniquely identifying the edges of a graph: The edge metric dimension

On the metric dimension of corona product graphs

On the strong metric dimension of corona product graphs and join graphs

The k-metric dimension of a graph

Mixed metric dimension of graphs