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This edition of IWONT is dedicated to Professor Miquel Àngel Fiol on the occasion of his 60th birthday

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### Summary

• #### Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010 ﻿

(2011-02)
Conference lecture
Open Access
• #### The beginnings ﻿

(2011)
Conference report
Open Access
The simple title tells many things. Of course, it could tell almost nothing since a well-known motto says that ‘everything must have a beginning’. However, another motto is more appropriate here: ‘A journey of a thousand ...
• #### Infinite families of 3-numerical semigroups with arithmetic-like links ﻿

(2011)
Conference report
Open Access
Let S = a, b,N be a numerical semigroup generated by a, b,N ∈ N with 1 < a < b < N and gcd(a, b,N) = 1. The conductor of S, denoted by c(S) or c(a, b,N), is the minimum element of S such that c(S) + m ∈ S for all m ∈ N ...
• #### On identifying codes in partial linear spaces ﻿

(2011)
Conference report
Open Access
Let (P,L, I) be a partial linear space and X ⊆ P ∪ L. Let us denote by (X)I = x∈X{y : yIx} and by [X] = (X)I ∪ X. With this terminology a partial linear space (P,L, I) is said to admit a (1,≤ k)-identifying code if the ...
• #### Subdivisions in a bipartite graph ﻿

(2011)
Conference report
Open Access
Given a bipartite graph G with m and n vertices, respectively,in its vertices classes, and given two integers s, t such that 2 ≤ s ≤ t, 0 ≤ m−s ≤ n−t, and m+n ≤ 2s+t−1, we prove that if G has at least mn−(2(m−s)+n−t) edges ...
• #### On the λ'-optimality of s-geodetic digraphs ﻿

(2011)
Conference report
Open Access
For a strongly connected digraph D the restricted arc-connectivity λ'(D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D − S has a non trivial strong component D1 such that D − V ...
• #### On the connectivity and restricted edge-connectivity of 3-arc graphs ﻿

(2011)
Conference report
Open Access
A 3−arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ←→G denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is ...
• #### Edge-superconnectivity of semiregular cages with odd girth ﻿

(2011)
Conference report
Open Access
A graph is said to be edge-superconnected if each minimum edge-cut consists of all the edges incident with some vertex of minimum degree. A graph G is said to be a {d, d + 1}- semiregular graph if all its vertices have ...
• #### M-Matrix Inverse problem for distance-regular graphs ﻿

(2011)
Conference report
Open Access
We analyze when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is a M–matrix;that is, it has non–positive off–diagonal elements or, equivalently when the Moore-Penrose inverse of the ...
• #### Graphs with equal domination and 2-domination numbers ﻿

(2011)
Conference report
Open Access
For a graph G a subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number γk(G) is the minimum cardinality among the k-dominating sets of G. Note ...
• #### Spectral behavior of some graph and digraph compositions ﻿

(2011)
Conference report
Open Access
Let G be a graph of order n the vertices of which are labeled from 1 to n and let $G_1$, · · · ,$G_n$ be n graphs. The graph composition G[$G_1$, · · · ,$G_n$] is the graph obtained by replacing the vertex i of G by the ...
• #### On the existence of combinatorial configurations ﻿

(2011)
Conference report
Open Access
A (v, b, r, k) combinatorial configuration can be defined as a connected, (r, k)-biregular bipartite graph with v vertices on one side and b vertices on the other and with no cycle of length 4. Combinatorial configurations ...
• #### Symmetric L-graphs ﻿

(2011)
Conference report
Open Access
In this paper we characterize symmetric L-graphs, which are either Kronecker products of two cycles or Gaussian graphs. Vertex symmetric networks have the property that the communication load is uniformly distributed on ...
• #### On the vulnerability of some families of graphs ﻿

(2011)
Conference report
Open Access
The toughness of a noncomplete graph G is defined as τ (G) = min{|S|/ω(G − S)}, where the minimum is taken over all cutsets S of vertices of G and ω(G − S) denotes the number of components of the resultant graph G − S by ...
• #### On the diameter of random planar graphs ﻿

(2011)
Conference report
Open Access
• #### An overview of the degree/diameter problem for directed, undirected and mixed graphs ﻿

(2011)
Conference report
Open Access
A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, ...
• #### Large graphs of diameter two and given degree ﻿

(2011)
Conference report
Open Access
Let r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these ...
• #### Fiedler’s Clustering on m-dimensional Lattice Graphs ﻿

(2011)
Conference report
Open Access
We consider the partitioning of m-dimensional lattice graphs using Fiedler’s approach , that requires the determination of the eigenvector belonging to the second smallest eigenvalue of the Laplacian. We examine the ...
• #### Large digraphs of given diameter and degree from coverings ﻿

(2011)
Conference report
Open Access
We show that a construction of Comellas and Fiol for large vertex-transitive digraphs of given degree and diameter from small digraphs preserves the properties of being a Cayley digraph and being a regular covering.
• #### Dual concepts of almost distance-regularity and the spectral excess theorem ﻿

(2011)
Conference report
Open Access
Generally speaking, ‘almost distance-regular’ graphs are graphs that share some, but not necessarily all, regularity properties that characterize distance-regular graphs. In this paper we first propose two dual concepts ...
• #### Large Edge-non-vulnerable Graphs ﻿

(2011)
Conference report
Open Access
In this paper we study the graphs such that the deletion of any edge does not increase the diameter. We give some upper bounds for the order of such a graph with given maximum degree and diameter. On the other hand ...
• #### A mathematical model for dynamic memory networks ﻿

(2011)
Conference report
Open Access
The aim of this paper is to bring together the work done several years ago by M.A. Fiol and the other authors to formulate a quite general mathematical model for a kind of permutation networks known as dynamic memories. A ...
• #### Algebraic characterizations of bipartite distance-regular graphs ﻿

(2011)
Conference report
Open Access
Bipartite graphs are combinatorial objects bearing some interesting symmetries. Thus, their spectra—eigenvalues of its adjacency matrix—are symmetric about zero, as the corresponding eigenvectors come into pairs. Moreover, ...
• #### Topology of Cayley graphs applied to inverse additive problems ﻿

(2011)
Conference report
Open Access
We present the basic isopermetric structure theory, obtaining some new simplified proofs. Let 1 ≤ r ≤ k be integers. As an application, we obtain simple descriptions for the subsets S of an abelian group with |kS| ≤ ...