# Extended Configurational Polyhedra Based on Graph Representation for Crystalline Solids

## TL;DR

This paper introduces an advanced graph-based method combined with the generalized Ising model to systematically identify extremal structures in crystalline solids, expanding the traditional configurational polyhedra to include more diverse and characteristic configurations.

## Contribution

It presents a novel approach that extends traditional configurational polyhedra using graph theory and GIM, enabling the inclusion of a broader set of extremal structures without prior interaction or element information.

## Key findings

- Constructed extended CP including additional topologically characteristic structures
- Demonstrated the method's ability to identify extremal structures more comprehensively
- Enhanced the understanding of configurational space in crystalline solids

## Abstract

We propose theoretical approach based on combination of graph theory and generalized Ising model (GIM), which enables systematic determination of extremal structures for crystalline solids without any information about interactions or constituent elements. The conventional approach to find such set of structure typically employs configurational polyhedra (CP) on configuration space based on GIM description, whose vertices can always be candidates to exhibit maximum or minimum physical quantities. We demonstrate that the present approach can construct extended CP whose vertices not only include those found in conventional CP, but also include other topologically and/or configurationally characteristic structures on the same dimensional configuration space with the same set of figures composed of underlying lattice points, which therefore has significant advantage over the conventional approach.