TETRANACCI GRAPHS

Abstract

Graphs are fundamental structures widely used across diverse scientific fields, including chemistry, biology, network theory, and the social sciences. In recent years, the study of graph-theoretic models involving specific integer sequences has attracted considerable interest, owing to their rich combinatorial and algebraic characteristics. Among these sequences, the Tetranacci numbers stand out for their notable connections to continued fractions, quadratic fields, and certain classes of Diophantine equations. In this study, we introduce a new family of graphs, termed “Tetranacci graphs”, whose degree sequences are composed of n consecutive Tetranacci numbers. Using the graph invariant Ω(D) — a tool that provides critical insights into the structural features of graphs such as realizability, connectivity, and cyclic components— we explore the conditions under which such sequences yield valid graphical realizations. We derive necessary and sufficient criteria for the realizability of these sequences for any positive integer n, and we classify all possible graphical forms (Tetranacci graphs) explicitly for the cases 1≤n≤4. Our analysis is further extended to offer a general structural characterization for the case n ≥5. These fundamental criteria establish a comprehensive framework for studying Tetranacci graphs, opening new avenues for research into the interplay between number theory and graph realizability.