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Moments in graphs
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Discrete applied mathematics
Date of publication: 2013
Journal article
Read the abstract Access to the full text Share Reference managersThis parameter generalizes, or it is closely related to, some wellknown graph invariants, such as the Wiener index W (G), when rho(u) = 1/2 for every u is an element of V, and the degree distance D'(G), obtained when rho(u) = delta(u), the degree of vertex u. In this paper we derive some exact formulas for computing the rhomoment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding rhomoments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same rhomoment for every rho (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product.
Let G be a connected graph with vertex set V and a weight function that assigns a nonnegative number to each of its vertices. Then, the moment of G at vertex u is de ned to be M G(u) = P v2V (v) dist(u; v), where dist( ; ) stands for the distance function. Adding up all these numbers, we obtain the moment of G: This parameter generalizes, or it is closely related to, some wellknown graph invari ants, such as the Wiener index W(G), when (u) = 1=2 for every u 2 V , and the degree distance D0(G), obtained when (u) = (u), the degree of vertex u. In this paper we derive some exact formulas for computing the moment of a graph obtained by a general operation called graft product, which can be seen as a generalization of the hierarchical product, in terms of the corresponding moments of its factors. As a consequence, we provide a method for obtaining nonisomorphic graphs with the same moment for every (and hence with equal mean distance, Wiener index, degree distance, etc.). In the case when the factors are trees and/or cycles, techniques from linear algebra allow us to give formulas for the degree distance of their product.
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Optimización y problemas extremales en teoria de grafos y combinatoria. Aplicacions a les redes de comunicación
Aguilo Gost, Francisco de Asis Luis; Abiad Monge, Aida; Andrés Yebra, José Luis; Barriere Figueroa, Eulalia; Cámara Vallejo, Marc; Ball, Simeon Michael; Comellas Padro, Francesc de Paula; Dalfo Simo, Cristina; Fàbrega Canudas, Josep; Garriga Valle, Ernest; Gomez Marti, Jose; Llado Sanchez, Anna; López Masip, Susana Clara; Miralles De La Asuncion, Alicia; Mitjana Riera, Margarida; Muñoz Lopez, Francisco Javier; Pelayo Melero, Ignacio Manuel; Pérez Mansilla, Sonia; Rius Font, Miquel; Serra Albo, Oriol; Vena, Lluis; Vilaltella Castanyer, Joan; Zaragoza Monroig, Maria Luisa; Espona Dones, Margarida; Sau Valls, Ignasi; Montejano Cantoral, Amanda; Perarnau Llobet, Guillem; Moragas Vilarnau, Jordi; Vena Cros, Lluís; Andres Yebra, Jose Luis; Fiol Mora, Miquel Àngel
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Edgedistanceregular graphs
Cámara Vallejo, Marc; Dalfo Simo, Cristina; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
European Conference on Combinatorics, Graph Theory and Applications
Presentation's date: 20110831
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Edgedistanceregular graphs
Cámara Vallejo, Marc; Dalfo Simo, Cristina; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Journal of combinatorial theory. Series A
Date of publication: 2011
Journal article
Read the abstract View Share Reference managersEdgedistanceregularity is a concept recently introduced by the authors which is similar to that of distanceregularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edgedistanceregular when it is distanceregular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edgedistanceregular if and only if its kincidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distanceregular graphs is proved, so giving a quasispectral characterization of edgedistanceregularity. Finally, it is shown that every nonbipartite graph which is both distanceregular and edgedistanceregular is a generalized odd graph. 
Dual concepts of almost distanceregularity and the spectral excess theorem
Dalfo Simo, Cristina; Van Dam, Edwin R; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Discrete mathematics
Date of publication: 2011
Journal article
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ALGEBRAIC AND COMBINATORIC APPROACH TO PSEUDODISTANCEREGULARITY OF GRAPHS AND COMPLETELY PSEUDOREGULAR CODES.
Cámara Vallejo, Marc
Defense's date: 20110615
Department of Applied Mathematics IV, Universitat Politècnica de Catalunya
Theses
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ALGEBRAIC AND COMBINATORIC APPROACH TO PSEUDODISTANCEREGULARITY OF GRAPHS AND COMPLETELY PSEUDOREGULAR CODES
Cámara Vallejo, Marc
Defense's date: 20110615
Department of Applied Mathematics IV, Universitat Politècnica de Catalunya
Theses
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A differential approach for bounding the index of graphs under perturbations
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Date: 201103
Report
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Edgedistanceregular graphs
Cámara Vallejo, Marc; Dalfo Simo, Cristina; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Date: 20110311
Report
Read the abstract Access to the full text Share Reference managersEdgedistanceregularity is a concept recently introduced by the authors which is similar to that of distanceregularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edgedistanceregular when it is distanceregular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edgedistanceregular if and only if its kincidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distanceregular graphs is proved, so giving a quasispectral characterization of edgedistanceregularity. Finally, it is shown that every nonbipartite graph which is both distanceregular and edgedistanceregular is a generalized odd graph. 
A differential approach for bounding the index of graphs under perturbations
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic journal of combinatorics
Date of publication: 20110902
Journal article
Read the abstract Access to the full text Share Reference managersThis paper presents bounds for the variation of the spectral radius (G) of a graph G after some perturbations or local vertex/edge modifications of G. The perturbations considered here are the connection of a new vertex with, say, g vertices of G, the addition of a pendant edge (the previous case with g = 1) and the addition of an edge. The method proposed here is based on continuous perturbations and the study of their differential inequalities associated. Within rather economical information (namely, the degrees of the vertices involved in the perturbation), the best possible inequalities are obtained. In addition, the cases when equalities are attained are characterized. The asymptotic behavior of the bounds obtained is also discussed. 
Edge distanceregular graphs
Cámara Vallejo, Marc; Dalfo Simo, Cristina; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic notes in discrete mathematics
Date of publication: 2011
Journal article
Read the abstract Access to the full text Share Reference managersEdgedistanceregularity is a concept recently introduced by the authors which is similar to that of distanceregularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edgedistanceregular when it is distanceregular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edgedistanceregular if and only if its kincidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distanceregular graphs is proved, so giving a quasispectral characterization of edgedistanceregularity. Finally, it is shown that every nonbipartite graph which is both distanceregular and edgedistanceregular is a generalized odd graph.
* distanceregularity; * local spectra; * predistance polynomials; * the spectral excess theorem; * generalized odd graphs 
When almost distance regularity attains distance regularity
Dalfo Simo, Cristina; Van Dam, Edwin R; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
International Workshop on Optimal Network Topologies
Presentation's date: 20100610
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Dual concepts of almost distanceregularity and the spectral excess theorem
Dalfo Simo, Cristina; Van Dam, Edwin R; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
International Workshop on Optimal Network Topologies
Presentation of work at congresses
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Combinatorial vs. Algebraic Characterizations of Completely PseudoRegular Codes
Cámara Vallejo, Marc; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic journal of combinatorics
Date of publication: 20100308
Journal article
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A simple proof of the spectral excess theorem for distanceregular graphs
Fiol Mora, Miquel Àngel; Gago Alvarez, Silvia; Garriga Valle, Ernest
Linear algebra and its applications
Date of publication: 20100415
Journal article
Read the abstract View Share Reference managersThe spectral excess theorem provides a quasispectral characterization for a (regular) graph Γ with d+1 distinct eigenvalues to be distanceregular graph, in terms of the excess (number of vertices at distance d) of each of its vertices. The original approach, due to Fiol and Garriga in 1997, was obtained by using a local approach, so giving a characterization of the socalled pseudodistanceregularity around a vertex. In this paper we present a new simple projection method based in a global point of view, and where the mean excess plays an essential role. 
The geometry of tspreads in kwalkregular graphs
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Journal of graph theory
Date of publication: 201008
Journal article
Read the abstract Access to the full text Share Reference managersA graph is walkregular if the number of closed walks of length rooted at a given vertex is a constant through all the vertices for all . For a walkregular graph G with d+1 different eigenvalues and spectrally maximum diameter D=d, we study the geometry of its dspreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a threedimensional case) and we compute its parameters. Moreover, the results are generalized to the case of kwalkregular graphs, a family which includes both walkregular and distanceregular graphs, and their tspreads or vertices at distance t from each other. 
On Almost DistanceRegular Graphs
Van Dam, Edwin R; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Gorissen, Bram
Date: 20100312
Report
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On almost distanceregular graphs
Dalfo Simo, Cristina; Van Dam, Edwin R; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Gorissen, Bram
Journal of combinatorial theory. Series A
Date of publication: 2010
Journal article
Read the abstract View Share Reference managersDistanceregular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distanceregular graphs, we study ‘almost distanceregular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distanceregular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walkregular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distanceregularity and walkregularity called mwalkregularity. Another studied concept is that of mpartial distanceregularity or, informally, distanceregularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distanceregularity, such as their common generalization of ( ,m)walkregularity. We introduce the concepts of punctual distanceregularity and punctual walkregularity as a fundament upon which almost distanceregular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distanceregular becomes whole distanceregular. We also give several characterizations of punctually distanceregular graphs that are generalizations of the spectral excess theorem. 
Characterizing (l, m)walkregular graphs
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Linear algebra and its applications
Date of publication: 20101230
Journal article
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Number of walks and degree powers in a graph
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Discrete mathematics
Date of publication: 200904
Journal article
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On kWalkRegular graphs
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic journal of combinatorics
Date of publication: 200904
Journal article
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Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes
Cámara Vallejo, Marc; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic journal of combinatorics
Date of publication: 200907
Journal article
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PROBLEMAS EXTREMALES Y DE OPTIMIZACIÓN EN TEORIA DE GRAFOS Y COMBINATORIA: APLICACIÓN AL ANALISIS Y ALGORITMOS DE REDES DE COMUNICAC
Abiad Monge, Aida; Andres Yebra, Jose Luis; Aguilo Gost, Francisco de Asis Luis; Aroca Farrerons, Josep Maria; Ball, Simeon Michael; Barajas Tomas, Javier; Barguilla Navarrete, Jorge; Barriere Figueroa, Eulalia; Cámara Vallejo, Marc; Comellas Padro, Francesc de Paula; Dalfo Simo, Cristina; Espona Dones, Margarida; Fàbrega Canudas, Josep; Gago Alvarez, Silvia; Garriga Valle, Ernest; Gomez Marti, Jose; Llado Sanchez, Anna; López Masip, Susana Clara; Miralles De La Asuncion, Alicia; Mitjana Riera, Margarida; Montejano Cantoral, Amanda; Moragas Vilarnau, Jordi; Muñoz Lopez, Francisco Javier; Pelayo Melero, Ignacio Manuel; Perarnau Llobet, Guillem; Pérez Mansilla, Sonia; Rius Font, Miquel; Sau Valls, Ignasi; Serra Albo, Oriol; Vena, Lluis; Vilaltella Castanyer, Joan; Zaragoza Monroig, Maria Luisa; Vena Cros, Lluís; Fiol Mora, Miquel Àngel
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COMBINATÒRIA , TEORIA DE GRAFS I APLICACIONS
Moragas Vilarnau, Jordi; Comellas Padro, Francesc de Paula; López Masip, Susana Clara; Mitjana Riera, Margarida; Llado Sanchez, Anna; Barriere Figueroa, Eulalia; Gomez Marti, Jose; Miralles De La Asuncion, Alicia; Pérez Mansilla, Sonia; Muñoz Lopez, Francisco Javier; Zaragoza Monroig, Maria Luisa; Dalfo Simo, Cristina; Garriga Valle, Ernest; Espona Dones, Margarida; Rius Font, Miquel; Aguilo Gost, Francisco de Asis Luis; Aroca Farrerons, Josep Maria; Cámara Vallejo, Marc; Gago Alvarez, Silvia; Ball, Simeon Michael; Andres Yebra, Jose Luis; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Pelayo Melero, Ignacio Manuel; Serra Albo, Oriol
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On kwalkregular graphs
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
British Combinatorial Conference
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On tcliques in kwalkregular graphs
Dalfo Simo, Cristina; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic notes in discrete mathematics
Date of publication: 200908
Journal article
Read the abstract View Share Reference managersA graph is walkregular if the number of cycles of length rooted at a given vertex is a constant through all the vertices. For a walkregular graph G with d+1 different eigenvalues and spectrally maximum diameter D = d, we study the geometry of its dcliques, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a regular tetrahedron and we compute its parameters. Moreover, the results are generalized to the case of kwalkregular graphs, a family which includes both walkregular and distanceregular graphs, and their tcliques or vertices at distance t from each other. 
Algunas aplicaciones de polinomios ortogonales de variable discreta a grafos y códigos
Cámara Vallejo, Marc; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Seminario de Matemática Discreta
Presentation's date: 20080909
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Algunas aplicaciones de polinomios octogonales de variable discreta a grafos y códigos
Cámara Vallejo, Marc; Fàbrega Canudas, Josep; Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Jornadas de Matemática Discreta y Algorítmica
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Algunas aplicaciones de polinomios ortogonales de variable discreta a grafos y códigos
Garriga Valle, Ernest
Seminario de Matemática Discreta
Presentation's date: 20080909
Presentation of work at congresses
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Spectral and Geometric Properties of kWalkRegular Graphs
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Electronic notes in discrete mathematics
Date of publication: 200708
Journal article
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Una experiencia docente orientada a incrementar el trabajo personal del estudiante
Otero Calviño, Beatriz; Martí Farré, Jaume; Garriga Valle, Ernest; Alonso Maleta, Maria Aranzazu; LLUIS, PRAT; Prat Viñas, Luis
Jornadas de Enseñanza Universitaria de la Informática
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On outindependent subgraphs of strongly regular graphs
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Linear and multilinear algebra
Date of publication: 200603
Journal article
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On the spectrum of an extremal graph with four eigenvalues
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Discrete mathematics
Date of publication: 200609
Journal article
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Number of Walks and Degree Powers in a Graph*
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Date: 200604
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Avoiding monocoloured triangles when colouring Kn
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Andres Yebra, Jose Luis
Date: 200408
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An algebraic characterization of completely regular codes in distanceregular graphs
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
SIAM journal on discrete mathematics
Date of publication: 200203
Journal article
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Pseudostrong regularity around a set
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Linear and multilinear algebra
Date of publication: 200203
Journal article
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Aprenentatge de càlcul1. successions, contiuïtat i derivació.
Aguilo Gost, Francisco de Asis Luis; Miralles De La Asuncion, Alicia; Garriga Valle, Ernest; Barguilla Navarrete, Jorge
Date of publication: 20020930
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Aprenentatge de Càlcul 2. Integració i sèries.
Aguilo Gost, Francisco de Asis Luis; Garriga Valle, Ernest; Miralles De La Asuncion, Alicia; Barguilla Navarrete, Jorge
Date of publication: 20020930
Book
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Algebraic characterizations of completely regular codes
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
SIAM journal on discrete mathematics
Date of publication: 200110
Journal article
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Boundary graphs: The limit case of a spectral property
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Andres Yebra, Jose Luis
Discrete mathematics
Date of publication: 200101
Journal article
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On twisted odd graphs
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Andres Yebra, Jose Luis
Combinatorics probability and computing
Date of publication: 200005
Journal article
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AN ALGEBRAIC CHARACTERIZATION OF COMPLETELY REGULAR CODES IN DISTANCEREGULAR GRAPHS. Codi Biblioteca 1400449878
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Date: 200008
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SOME APPLICATIONS OF ORTHOGONAL POLYNOMIALS OF A DISCRETE VARIABLE TO GRAPHS AND CODES. Codi Biblioteca 1400449942
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Date: 200007
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A LOCAL APPROACH TO THE STUDY OF DISTANCERELATED PARAMETERS OF A GRAPH FROM ITS SPECTRUM. Codi Biblioteca 1400449941
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Date: 200001
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On the algebraic theory of pseudodistanceregulariry around a set
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Linear algebra and its applications
Date of publication: 199909
Journal article
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A DIFFERENTIAL APPROACH TO FIND BOUNDS OF THE SPECTRAL RADIUS OF GRAPHS UNDER PERTURBACIONS. Codi Biblioteca 1400449937
Garriga Valle, Ernest
Date: 199911
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Boundary graphsII: The limit case of a spectral property
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Andres Yebra, Jose Luis
Discrete mathematics
Date of publication: 199801
Journal article
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From regular boundary graphs to antipodal distanceregular graphs
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest; Andres Yebra, Jose Luis
Journal of graph theory
Date of publication: 199801
Journal article
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The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs
Fiol Mora, Miquel Àngel; Garriga Valle, Ernest
Discrete applied mathematics
Date of publication: 199810
Journal article
Read the abstract Access to the full text Share Reference managersLet Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ_1 > λ_2 > · · · > λ_d. For every k = 0,1, . . . ,d −1, the kalternating polynomial P_k is defined to be the polynomial of degree k and norm 
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