# Smith and Critical groups of Polar Graphs

## TL;DR

This paper calculates the elementary divisors of adjacency and Laplacian matrices for polar graphs, which are based on isotropic subspaces in finite vector spaces, providing insights into their algebraic structure.

## Contribution

It introduces a method to compute elementary divisors for polar graphs' matrices, advancing understanding of their algebraic properties.

## Key findings

- Elementary divisors of adjacency matrices determined
- Elementary divisors of Laplacian matrices determined
- Enhanced understanding of polar graphs' algebraic structure

## Abstract

We compute the elementary divisors of the adjacency and Laplacian matrices of families of polar graphs. These graphs have as vertices the isotropic one-dimensional subspaces of finite vector spaces with respect to non-degenerate forms, with adjacency given by orthogonality.