Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts.

We propose an analysis of the codified Law of France as a structured system. Fifty two legal codes are selected on the basis of explicit legal criteria and considered as vertices with their mutual quotations forming the edges in a network which properties are analyzed relying on graph theory. We find that a group of 10 codes are simultaneously the most citing and the most cited by other codes, and are also strongly connected together so forming a "rich club" sub-graph. Three other code communities are also found that somewhat partition the legal field is distinct thematic sub-domains. The legal interpretation of this partition is opening new untraditional lines of research. We also conjecture that many legal systems are forming such new kind of networks that share some properties in common with small worlds but are far denser. We propose to call "concentrated world".

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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. A private communication of Behruz Tayfeh-Rezaie pointed out that an even lollipop with a cycle of length at least $6$ is determined by its spectrum but the result for lollipops with a cycle of length $4$ is still unknown. We give an unified proof for lollipops with a cycle of length not equal to $4$, generalize it for lollipops with a cycle of length $4$ and therefore answer the question. Our proof is essentially based on a method of counting closed walks.

We explore one aspect of the structure of a codified legal system at the national level using a new type of representation to understand the strong or weak dependencies between the various fields of law. In Part I of this study, we analyze the graph associated with the network in which each French legal code is a vertex and an edge is produced between two vertices when a code cites another code at least one time. We show that this network distinguishes from many other real networks from a high density, giving it a particular structure that we call concentrated world and that differentiates a national legal system (as considered with a resolution at the code level) from small-world graphs identified in many social networks. Our analysis then shows that a few communities (groups of highly wired vertices) of codes covering large domains of regulation are structuring the whole system. Indeed we mainly find a central group of influent codes, a group of codes related to social issues and a group of codes dealing with territories and natural resources. The study of this codified legal system is also of interest in the field of the analysis of real networks. In particular we examine the impact of the high density on the structural characteristics of the graph and on the ways communities are searched for. Finally we provide an original visualization of this graph on an hemicyle-like plot, this representation being based on a statistical reduction of dissimilarity measures between vertices.In Part II (a following paper) we show how the consideration of the weights attributed to each edge in the network in proportion to the number of citations between two vertices (codes) allows deepening the analysis of the French legal system.

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian ([2]) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the s-homotopy and the graph homotopy defined in [5].

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. To cite this article: R. Boulet, C. R. Acad. Sci. Paris, Ser. I 346 (2008). RésuméLe mille-pattes est déterminé par le spectre du Laplacien. Un mille-pattes est un graphe obtenu en attachant un sommet pendantà chaque sommet de degré 2 d'une chaîne. Dans cet article nous montrons qu'un mille-pattes est déterminé par le spectre du Laplacien. Pour citer cet article : R. Boulet, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Version française abrégéeLe Laplacien L d'un graphe est la matrice L = D − A où D est la matrice diagonale des degrés et A est la matrice d'adjacence du graphe. Le spectre du Laplacien donne des informations sur la structure du graphe, comme la connexité (voir [2, 4] pour plus de détails), ces informations sont souvent insuffisantes pour reconstruire le grapheà partir du spectre et la question « Quels graphes sont déterminés par leur spectre ? » [2] demeure un problème difficile. En particulier il est connu [5] que presque aucun arbre n'est déterminé par le spectre du Laplacien et seules quelques familles d'arbres déterminés par leur spectre ont jusqu'alorsété découvertes (citons par exemple [6] et [7]).On appelle mille-pattes le graphe obtenu en attachant un sommet pendantà chaque sommet de degré 2 d'une chaîne (voir figure 1). Nous montrons dans cet 1 article que le mille-pattes est déterminé par le spectre du Laplacien, enrichissant ainsi les familles connues d'arbres déterminés par le spectre du Laplacien.On note P k la chaîneà k sommets et T le triangle. Deux graphes sont dits A-cospectraux (resp.

Abstract.We perform a detailed analysis of the network constituted by the citations in a legal code, we search for hidden structures and properties. The graph associated to the Environmental code has a small-world structure and it is partitioned in several hidden communities of articles that only partially coincide with the organization of the code as given by its table of content. Several articles are also connected with a low number of articles but are intermediate between large communities. The structure of the Environmental Code is contrasting with the reference network of all the French Legal Codes that presents a rich-club of ten codes very central to the whole French legal system, but no small-world property. This comparison shows that the structural properties of the reference network associated to a legal system strongly depends on the scale and granularity of the analysis, as is the case for many complex systems.