Derivation of closed-form analytical irregularity topological indices for ideal rock-salt lattices

Abstract

Magnesium oxide (MgO) is a prototypical ionic solid with a well-defined rock-salt lattice and is frequently used as a reference structure in materials science and theoretical modeling. In this work, we present a rigorous graph-theoretical characterization of an idealized MgO lattice, formulated as a periodic point-lattice for analytical purposes. The structure is treated as a two-dimensional topological projection of the three-dimensional rock-salt lattice, enabling exact symbolic analysis without incorporating surface relaxation, reconstruction, or energetic considerations. Using edge partitioning and degree-based methods, we derive closed-form analytical expressions for a class of irregularity topological indices (ITIs) associated with the resulting periodic graph. These indices quantify degree asymmetry and structural heterogeneity purely in a combinatorial sense and serve as mathematical descriptors of connectivity patterns in finite lattice graphs. The primary contribution of this study is the exact derivation of these irregularity indices for an ideal rock-salt lattice, which, to the best of our knowledge, has not been previously reported. It is emphasized that the present work is formulated entirely within a graph-theoretical framework and does not establish direct correlations with experimentally measured physical properties. The observed scaling behavior of the indices reflects topological boundary effects inherent to finite periodic graphs and should not be interpreted as physical disorder in real crystalline materials. Instead, the results provide a foundational mathematical framework for future studies on non-ideal lattices, such as defective, doped, or reconstructed systems, where irregularity-based descriptors may be meaningfully related to material properties.