Papers by Margarida Mitjana
Linear Algebra and its Applications, 2019
In this paper we introduce new effective resistances on a given network, associated with a positi... more In this paper we introduce new effective resistances on a given network, associated with a positive value and two weights on the vertex set and investigate under which conditions they determine a distance. We prove that this property is closely related with superharmonicity. Moreover, we analyze the behavior of these distances under the usual network transformations, specially the so-called star-mesh transformation. We also compute the effective resistance for an extended network; that is the network obtained from the former one by joining a new vertex, and then study the effect of the contraction of this new network; that is we apply a star-mesh transformation with center in the joined vertex.
arXiv: Combinatorics, Aug 11, 2016
We study a family of graphs related to the n-cube. The middle cube graph of parameter k is the su... more We study a family of graphs related to the n-cube. The middle cube graph of parameter k is the subgraph of Q 2k−1 induced by the set of vertices whose binary representation has either k − 1 or k number of ones. The middle cube graphs can be obtained from the wellknown odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).
arXiv: Classical Analysis and ODEs, May 28, 2015
We present here necessary and sufficient conditions for the invertibility of circulant and symmet... more We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the solution of boundary value problems associated to second order linear difference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coefficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.
Advanced Courses in Mathematics - CRM Barcelona, 2018
This book contains the notes of the lectures delivered at an Advanced Course on Combinatorial Mat... more This book contains the notes of the lectures delivered at an Advanced Course on Combinatorial Matrix Theory held at Centre de Recerca Matemàtica (CRM) in Barcelona. These notes correspond to five series of lectures. The first series is dedicated to the study of several matrix classes defined combinatorially, and was delivered by Richard A. Brualdi. The second one, given by Pauline van den Driessche, is concerned with the study of spectral properties of matrices with a given sign pattern. Dragan Stevanović delivered the third one, devoted to describing the spectral radius of a graph as a tool to provide bounds of parameters related with properties of a graph. The fourth lecture was delivered by Stephen Kirkland and is dedicated to the applications of the Group Inverse of the Laplacian matrix. The last one, given by Ángeles Carmona, focuses on boundary value problems on finite networks with special in-depth on the M-matrix inverse problem
Linear Algebra and its Applications, 2016
Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions as... more Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schrödinger operators on networks that have been obtained by adding a new vertex to a given network, can be seen as perturbations of the Schrödinger operators on the initial network. Therefore, the Green function on the new network can be computed in terms of the Green function of the original network.
En este trabajo presentamos un procedimiento de diseminación de la información en una red general... more En este trabajo presentamos un procedimiento de diseminación de la información en una red general. El criterio utilizado para hacer eficiente la elección de las ramas del emparejamiento en cada etapa del proceso consiste en asociar a cada rama un peso determinado por la medida de equilibrio del subconjunto de vértices no informados, aprovechando que esta medida tiene información sobre la conectividad de este subconjunto con su complementario. La aplicación del algoritmo al caso de algunas familias de redes para las que ya se conocen resultados confirma la bondad del nuevo método. * Trabajo parcialmente subvencionado por el Ministerio de Ciencia y Tecnología mediante el proyecto BFM2003-06014 y por la ETSECCPB.
The two main characteristics of small-world networks are strong local clustering, and small diame... more The two main characteristics of small-world networks are strong local clustering, and small diameter. These two characteristics are desirable properties in communica- tion networks since typical communication patterns show large amounts of local com- munication and a small amount of non-local communication that must be completed quickly. In this paper, we study variants of broadcasting that resemble the spread of computer viruses in networks. We also study the related problem of stopping an epidemic.
By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general ... more By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M–matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M-matrix of order two and three. We also consider the case of the cycle C4 an example of a non-contractible situation topologically different from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacc...
Electronic Notes in Discrete Mathematics, 2014
ABSTRACT
The local spectrum of a graph G = (V,E), constituted by the standard eigenvalues of G and their l... more The local spectrum of a graph G = (V,E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is “seen ” from a given vertex. Thus, for each vertex i ∈ V, the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λl ∈ evG is the sum, extended to all vertices, of its local multiplicities. In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the (regular o semiregular) graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results related to the number of cycles.
We consider the problem of routing uniform communication instances in switched optical tori that ... more We consider the problem of routing uniform communication instances in switched optical tori that use the wavelength division multiplexing (or WDM) approach. A communication instance is called uniform if it consists exactly of all pairs of nodes in the graph whose distance is equal to one from a specified set S = {dl, d2,..., d}. We give bounds on the optimal load induced on an edge for any uniform instance in a torus T,x,. When k = 1, we prove necessary and sufficient conditions on the value in S relative to n for the wavelength index to be equal to the load. When k _ 2, we show that for any set S, there exists an no, such that for all n no, there is an optimal wavelength assignment for the communication instance specified by S on the torus T,x,. We also show an approximation for the wavelength index for any S and n. Finally, we give some results for rectangular tori.
This paper is devoted to the gossip (or all-to-all) problem in the chordal ring under the one-por... more This paper is devoted to the gossip (or all-to-all) problem in the chordal ring under the one-port line model. The line model assumes long distance calls between non neighboring processors. In this sense, the line model is strongly related to circuit-switched networks, wormhole routing, optical networks supporting wavelength division multiplexing, ATM switching, and networks supporting connected mode routing protocols. Since the chordal rings are competitors of networks as meshes or tori because of theirs short diameter and bounded degree, it is of interest to ask whether they can support intensive communications (typically all-to-all) as e ciently as these networks. We propose polynomial algorithms to derive optimal or near optimal gossip protocols in the chordal ring.
Applicable Analysis and Discrete Mathematics
In this paper we consider a subdivision of a given network and we show how the group inverse matr... more In this paper we consider a subdivision of a given network and we show how the group inverse matrix of the normalized laplacian of the subdivision network is related to the group inverse matrix of the normalized laplacian of the initial given network. Our approach establishes a relationship between solutions of related Poisson problems on both structures and takes advantage on the properties of the group inverse matrix. As a consequence we get formulae for effective resistances and the Kirchhoff Index of the subdivision network expressed in terms of its corresponding in the base network. Finally, we study two examples where the base network are the star and the wheel, respectively.
The two main characteristics of small-world networks are strong local clustering, and small diame... more The two main characteristics of small-world networks are strong local clustering, and small diameter. These two characteristics are desirable properties in communication networks since typical communication patterns show large amounts of local communication and a small amount of non-local communication that must be completed quickly. In this paper, we study variants of broadcasting that resemble the spread of computer viruses in networks. Our deterministic results exhibit rates of “infection” that are similar to previously obtained probabilistic results for the spread of contagious diseases in populations.
Very often, strongly regular graphs appear associated with partial geometries. The point graph of... more Very often, strongly regular graphs appear associated with partial geometries. The point graph of a partial geometry is the graph whose vertices are the points of the geometry and adjacency is defined by collinearity. It is well known that the point graph associated to a partial geometry is a strongly regular graph and, in this case, the strongly regular graph is named geometric. When the parameters of a strongly regular graph, Γ, satisfy the relations of a geometric graph, then Γ is named a pseudo geometric graph. Moreover, it is known that not every pseudo geometric graph is geometric. In this work, we characterize strongly regular graphs that are pseudo geometric and we analyze when the complement of a pseudo geometric graph is also pseudo geometric.
Ars Comb., 2001
Cycle prefix digraphs are a class of Cayley coset graphs with many remarkable properties such as ... more Cycle prefix digraphs are a class of Cayley coset graphs with many remarkable properties such as symmetry, large number of nodes for a given degree and diameter, simple shortest path routing, Hamiltonicity, optimal connectivity, and others. In this paper we show that the cycle prefix digraphs, like the Kautz digraphs, contain cycles of all lengths l, with l between two and N, the order of the digraph, except for N 1.
In this paper we consider a subdivision of a given network and we show how the group inverse matr... more In this paper we consider a subdivision of a given network and we show how the group inverse matrix of the normalized laplacian of the subdivision network is related to the group inverse matrix of the normalized laplacian of the initial given network. Our approach establishes a relationship between solutions of related Poisson problems on both structures and takes advantage on the properties of the group inverse matrix. As a consequence we get formulae for effective resistances and the Kirchhoff Index of the subdivision network expressed in terms of its corresponding in the base network. Finally, we study two examples where the base network are the star and the wheel, respectively.
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Papers by Margarida Mitjana