# Orthogonal graphs modulo power of 2

## TL;DR

This paper introduces a new class of orthogonal graphs over modular integers of powers of two, classifies their regularity, computes parameters, and analyzes their symmetry, automorphisms, and coloring properties.

## Contribution

It defines and studies orthogonal graphs over rac{2^n}{} for the first time, classifying their regularity and determining their automorphism groups and chromatic numbers.

## Key findings

- The graphs are strongly regular or quasi-strongly regular with explicitly computed parameters.
- The graphs are arc transitive, indicating high symmetry.
- The chromatic number is determined except in specific cases.

## Abstract

In this work, we define an orthogonal graph on the set of equivalence classes of $(2\nu + \delta)-$tuples over $\mathbb{Z}_{2^n}$ where $n$ and $\nu$ are positive integers and $\delta = 0, 1$ or $2$. We classify our graph if it is strongly regular or quasi-strongly regular and compute all parameters precisely. We show that our graph is arc transitive. The automorphisms group is given and the chromatic number of the graph except when $\delta = 0$ and $\nu$ is odd is determined. Moreover, we work on subconstituents of this orthogonal graph.