Conference abstracts
Session S01 - Computational Algebra and Applications of Algebra
July 29, 18:00 ~ 18:25
Arithmetical structures of graphs.
Carlos E. Valencia Oleta
Mathematics department, Cinvestav-IPN, México - cvalencia@math.cinvestav.edu.mx
Given a graph $G=(V,E)$, its generalized Laplacian matrix is the square matrix given by \[ L(G,X_G)_{u,v}= \begin{cases} x_u&\text{if }u=v,\\ -m_{uv}&\text{if }u\neq v, \end{cases} \] where $X_G=\{x_u|u\in V(G)\}$ is the set of indeterminates indexed by the vertices of $G$ and $m_{uv}$ is the number of edges between $u$ and $v$. For any ${\bf d}\in \mathbb{N}^{V}$, let $L(G,{\bf d})$ be the integer matrix that result by making $x_u={\bf d}_u$ on $L(G,X_G)$. An arithmetical graph is a triplet $(G,{\bf d},{\bf r})$ given by a multidigraph $G$ and a pair of vectors $({\bf d},{\bf r})\in \mathbb{N}_+^V\times \mathbb{N}_+^V$ such that $\mathrm{gcd}({\bf r}_v\, | \,v\in V(G))=1$ and \[ L(G,{\bf d}){\bf r}^t={\bf 0}^t. \] Given an arithmetical graph $(G,{\bf d},{\bf r})$ we say that the pair $({\bf d},{\bf r})$ is an arithmetical structure of $G$. The concept of arithmetical graphs was introduced by Lorenzini as some intersection matrices that arise in the study of degenerating curves in algebraic geometry.
Under certain hypothesis, it can be prove that the number of arithmetical structures of $G$ is finite. In this way, let \[ \mathcal{A}(G)=\{({\bf d},{\bf r})\in\mathbb{N}_+^{V(G)}\times \mathbb{N}_+^{V(G)} \,| \,({\bf d},{\bf r})\textrm{ is an arithmetical structure of } G\}. \] In this talk we present a survey of the recent results obtained on arithmetical graphs. For instance we present how behaves $\mathcal{A}(G)$ under some operations of graphs like subdivision of edges, duplication of vertices, etc.
These results allows to compute $\mathcal{A}(G)$ for some families of graphs. For instance it can be prove that the number of arithmetical structures of a path $P_n$ with $n$ vertices is equal to the Catalan number $C_{n-1}$.