Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 72 – 92. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-1-72-92
Research Article
Full text in English
MSC: 05C05, 05C12, 05C20, 05C25, 05C35, 05C76, 68R10

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On Irregularity Measures in Trees with Fibonacci Degree Sequences

J. Hamoud¹, A.Ya. Belov¹, D.Abdullah^{\ast}¹, M. Almahalebi²

¹Moscow Institute of Physics and Technology, Moscow Institutskii per. 9, Dolgoprudnyi, 141700 Russia

²University of Ibn Tofail, Kenitra, Campus Universitaire, B.P. 242 14000, Morocco

Abstract. In this paper, degree-based topological indices are fundamental graph invariants used to quantify structural irregularity. Let G = (V, E) be a simple tree with vertex degrees dv for  v \in V(G). This paper investigates trees whose degree sequences are governed by the Fibonacci numbers \{F_k\}_{k \geq 3}, referred to here as Fibonacci trees. Particular attention is given to the Albertson index and the Sigma index \sigma(G) =\sum\limits_{uv \in E(G)} (du − dv)^2, both of which measure the degree irregularity along the edges of the graph. Firstly, characterize the Fibonacci degree sequences that are realizable by trees and describe their key structural properties. Exploiting the recursive nature of the Fibonacci sequence, we derive an explicit closed-form expression for the Albertson index of Fibonacci trees in terms of F_k. This formula reveals how recursive degree constraints influence edgedegree imbalances. Sharp lower and upper bounds for both irr(G) and \sigma(G) are then established for trees of fixed order with prescribed degree sequences. The extremal trees achieving these bounds are identified and include paths, stars, and certain star-like configurations. These results extend known extremal properties of degree-based topological indices and illustrate the specific impact of Fibonacci degree sequences on tree irregularity. The study offers new insights into recursively constrained degree sequences and suggests promising directions for further analysis of structured trees using algebraic and combinatorial methods.

Key words: trees, sequence, Zagreb, topological, Fibonacci, isomorphic.

Received: 06.02.2026; Revised: 16.02.2026; Accepted: 23.03.2026; First online: 29.03.2026

For citation. Hamoud J., Belov A.Ya., Abdullah D., Almahalebi M. On irregularity measures in trees with Fibonacci degree sequences. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 72-92. EDN: YJJZFW. https://doi.org/10.26117/2079-6641-2026-54-1-72-92.

Funding. Not funding

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: abdulla.d@phystech.edu

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Hamoud J., Belov A.Ya., Abdullah D., Almahalebi M., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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Information about the authors

Hamoud Jasem – PhD student, Moscow Institute of Physics and Technology, Russia, ORCID 0009-0002-0192-3627

Belov Alexey Yakovlevich – DSc (Phys. & Math.), Professor, Сhief Researcher, Moscow Institute of Physics and Technology, Russia, ORCID 0000-0002-1371-7479.

Abdullah Duaa – PhD student, Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia, ORCID 0009-0008-6855-1729.

Almahalebi Muaadh – PhD (Phys. & Math.), Professor, Сhief Researcher, Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco, ORCID 0000-0003-0647-8286.

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