The lollipop graph is determined by its spectrum

An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. We revisit the proof for odd lollipop, generalize it for even lollipop and therefore answer the question. Our proof is essentially based on a method of counting closed walks

Disjoint unions of complete graphs characterized by their Laplacian spectrum

International audienceA disjoint union of complete graphs is in general not determined by its Laplacian spectrum. We show in this paper that if we only consider the family of graphs without isolated vertex then a disjoint union of complete graphs is determined by its Laplacian spectrum within this family. Moreover we show that the disjoint union of two complete graphs with $a$ and $b$ vertices, $\frac{a}{b}>\frac{5}{3}$ and $b>1$ is determined by its Laplacian spectrum. A counter-example is given when $\frac{a}{b}=\frac{5}{3}$

Spectral behavior of some graph and digraph compositions

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 graph Gi and there is an edge between u ∈ $G_i$ and v ∈ $G_j$ if and only if there is an edge between i and j in G. We first consider graph composition G[$K_k$, · · · ,$K_k$] where G is regular and $K_k$ is a complete graph and we establish some links between the spectral characterisation of G and the spectral characterisation of G[$K_k$, · · · ,$K_k$]. We then prove that two non isomorphic graphs G[$G_1$, · · ·$G_n$] where $G_i$ are complete graphs and G is a strict threshold graph or a star are not Laplacian-cospectral, giving rise to a spectral characterization of these graphs. We also consider directed graphs, especially the vertex-critical tournaments without non-trivial acyclic interval which are tournaments of the shape t[$\overrightarrow{C}_{k_1}$, · · · ,$\overrightarrow{C}_{k_m}$], where t is a tournament and $\overrightarrow{C}_{k_i}$ is a circulant tournament. We give conditions to characterise these graphs by their spectrum.Peer Reviewe

Spectral characterizations of sun graphs and broken sun graphs

Graphs and AlgorithmsInternational audienceSeveral matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns

The centipede is determined by its Laplacian spectrum

A centipede is a graph obtained by appending a pendant vertex to each vertex of degree 2 of a path. In this paper we prove that the centipede is determined by its Laplacian spectrum.Comment: Article en anglais avec un r\'esum\'e \'etendu en fran\c{c}ais. Paper in english with an extended abstract in frenc

Clustering a medieval social network by SOM using a kernel based distance measure

6 pagesInternational audienceIn order to explore the social organization of a medieval peasant community before the Hundred Years' War, we propose the use of an adaptation of the well-known Kohonen Self Organizing Map to dissimilarity data. In this paper, the algorithm is used with a distance based on a kernel which allows the choice of a smoothing parameter to control the importance of local or global proximities

Partitionnement d'un réseau de sociabilité à fort coefficient de clustering

Article court, 6 pages.In order to compare social organization of a medieval peasantry before and after the Hundred Years' War we study the sructure of social networks built from a corpus of agrarian contracts. Low diameters and high clusterings show small-world graphs. Like many other networks studied these last years these graphs are scale-free. The distributions of the vertex degrees are fitted by a truncated power law. Moreover they have a rich-club : a dense core with a low diameter consisting of vertices with high degree. The particular shape of the laplacian spectrum allows us to extract communities that are spread along a star whose center is the rich-club

A New Influence Measure Based on Graph Centralities and Social Network Behavior Applied to Twitter Data

In this paper, we use graph theory to explore concepts of influence in socialized groups. When analyzing social networks, centrality indicators make it possible to assess the power of an individual. We discuss various centrality indicators and focus on degree and betweenness. After observing a strong correlation between them, we propose defining new assessments based on a decorrelation method that we characterize from different mathematical perspectives (algebraic, probabilistic, and statistical). We apply this theoretical framework to a network of tweets about the Uber versus taxi conflict, which took place in June, 2015, and for which we detected different influential individuals