Date of Defense
11-11-2025 11:00 AM
Location
F3-223
Document Type
Dissertation Defense
Degree Name
Doctor of Philosophy in Mathematics
College
COS
Department
Mathematical Sciences
First Advisor
Prof. Nafaa Chbili
Keywords
spectral graph theory; energy; topological indices; inverse sum indeg index; Sombor index; graph operations; zero-divisor graphs; distance energy; distance spectrum; commutative rings; QSPR.
Abstract
Spectral graph theory is a subfield of algebraic graph theory that studies the matrices associated with graphs. It lives at the nexus of Linear Algebra and Combinatorics. Many intriguing results in the domains of Matrix Theory and Combinatorics have come from studying the eigenvalues of graph matrices; in fact, several open problems in both areas have been resolved. Beyond its theoretical appeal, spectral graph theory has found meaningful applications in theoretical chemistry, particularly in the mathematical classification of chemical graphs. These classifications underpin quantitative structure–property relationships (QSPRs), facilitating the prediction of physicochemical properties such as enthalpy of vaporization, molar refractivity, and boiling point.
This dissertation explores the spectral properties of graph-associated matrices, particularly the inverse sum indeg (ISI), Sombor, and distance matrices. It identifies extremal graphs with respect to their spectral radius and energy, and characterizes graphs having a small number of distinct eigenvalues. New bounds for the ISI and Sombor indices are established for standard graph operations, including the corona, Cartesian, strong, composition, and join of graphs. The study further investigates distance spectra and energies of zero-divisor graphs derived from selected commutative rings, revealing precise structural relationships. Finally, the dissertation connects theoretical findings to chemical graph theory through a QSPR analysis, demonstrating meaningful correlations between ISI-based indices and physicochemical properties of chemical compounds.
Included in
PROBLEMS IN EXTREMAL GRAPH THEORY AND SPECTRAL GRAPH THEORY
F3-223
Spectral graph theory is a subfield of algebraic graph theory that studies the matrices associated with graphs. It lives at the nexus of Linear Algebra and Combinatorics. Many intriguing results in the domains of Matrix Theory and Combinatorics have come from studying the eigenvalues of graph matrices; in fact, several open problems in both areas have been resolved. Beyond its theoretical appeal, spectral graph theory has found meaningful applications in theoretical chemistry, particularly in the mathematical classification of chemical graphs. These classifications underpin quantitative structure–property relationships (QSPRs), facilitating the prediction of physicochemical properties such as enthalpy of vaporization, molar refractivity, and boiling point.
This dissertation explores the spectral properties of graph-associated matrices, particularly the inverse sum indeg (ISI), Sombor, and distance matrices. It identifies extremal graphs with respect to their spectral radius and energy, and characterizes graphs having a small number of distinct eigenvalues. New bounds for the ISI and Sombor indices are established for standard graph operations, including the corona, Cartesian, strong, composition, and join of graphs. The study further investigates distance spectra and energies of zero-divisor graphs derived from selected commutative rings, revealing precise structural relationships. Finally, the dissertation connects theoretical findings to chemical graph theory through a QSPR analysis, demonstrating meaningful correlations between ISI-based indices and physicochemical properties of chemical compounds.