A general method to obtain the spectrum and local spectra of a graph from its regular partitions

It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the C-local multiplicities of any class of vertices C is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.Partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Governmentunder projects PGC2018-095471-B-I00 and MTM2017-83271-R. Also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement no. 73492

A new approach to gross error detection for GPS networks

We present a new matrix-based approach to detect and correct gross errors in GPS geodetic control networks. The study is carried out by introducing a new matrix, whose entries are powers of a (real or complex) variable, which fully represents the network.This research have been partially supported by the Catalan Research Council, AGAUR, under project 2017SGR1087. The first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922

On d-Fibonacci digraphs

The d-Fibonacci digraphs F(d, k), introduced here, have the number of vertices following some generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, F(d, k) has diameter d + k − 2 and is semi-pancyclic; that is, it has a cycle of every length between 1 and ℓ, with ℓ ∈ {2k − 2, 2k − 1}. Moreover, it turns out that several other numbers of F(d, k) (of closed l-walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the d-Fibonacci digraphs.The research of the first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922

A note on the order of iterated line digraphs

Given a digraph G, we propose a new method to find therecurrence equation for the number of vertices nk of the k-iterated linedigraph Lk(G), for k≥0, where L0(G)=G. We obtain this result by usingthe minimal polynomial of a quotient digraph π(G) of GThis research is supported by the Ministerio de Economía y Competitividad and the European Regional Development Fund under project MTM2014-60127-P, and the Catalan Research Council under project 2014SGR1147

The spectral excess theorem for graphs with few eigenvalues whose distance- 2 or distance-1-or-2 graph is strongly regular

We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d+1 is the number of different eigenvalues of Γ. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.Research of C. Dalfó and M. A. Fiol is partially supported by Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR) under project 2017SGR1087. Research of J. Koolen is partially supported by the National Natural Science Foundation of China under project No. 11471009, and the Chinese Academy of Sciences under its ‘100 talent’ programme. The research of C. Dalfó has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922