Monojenik yarıgrup graflarının dereceye dayalı topolojik indisleri: İndirgenmiş Zagreb indisleri, koindisler ve PI indisi

Abstract

Topolojik indisler, graf yapılarının ve bunların nicel özelliklerinin incelenmesinde temel uygulamalarda önemli bir yere sahiptir. Bu çalışmada, saf ve uygulamalı matematik açısından da önem taşıyan, iyi bilinen bir cebirsel graf sınıfı olan monojenik yarıgrup graflarının indirgenmiş Zagreb indisini, Zagreb koindislerini ve PI indisini inceliyoruz. Grafın komşuluk yapısını temel araç olarak kullanarak bu indisler için tam kapalı form ifadeler elde ediyor ve sonuçları açık örneklerle gösteriyoruz. Ayrıca, monojenik yarıgrup graflarının indirgenmiş birinci Zagreb indisi, indirgenmiş ikinci Zagreb indisi, Zagreb koindisleri ve PI indisi için n'nin tek veya çift olmasına bağlı açık kapalı formüller elde ediyoruz. Bu türetimler, kenar kümesinin komşuluk-blok ayrışımına dayanmaktadır ve çeşitli derece-tabanlı topolojik indis aileleri için birleşik bir hesaplama çerçevesi sunmaktadır. Elde edilen sonuçlar, bir monojenik yarıgrubun cebirsel yapısının ilişkili grafın önemli kombinatoryal ve topolojik özelliklerini nasıl belirlediğini ortaya koymaktadır.

Keywords

• Monojenik yarıgruplar
• Zagreb indisleri
• PI indisi

Supporting Institution

Bu çalışma herhangi bir kurum, kuruluş veya proje tarafından desteklenmemiştir.

Ethical Statement

Bu çalışma tamamen teorik bir matematik çalışması olup insan veya hayvan deneği, kişisel veri, anket, mülakat veya deneysel uygulama içermemektedir. Bu nedenle etik kurul onayı gerektirmemektedir.

Thanks

Bu çalışmada teşekkür gerektiren herhangi bir özel katkı bulunmamaktadır.

Degree-based topological indices of monogenic semigroup graphs: reduced Zagreb indices, coindices and PI index

Abstract

Topological indices play a fundamental role in the study of graph structures and their quantitative properties, particularly in applications related to chemistry, network science, and dynamic system analysis. In this work, we study the Reduced Zagreb index, Zagreb coindices, and the PI index of monogenic semigroup graphs, a well-studied class of algebraic graphs with relevance to both pure and applied mathematics. Using the adjacency structure of the graph as the main tool, we derive exact closed-form expressions for these indices and illustrate the results through explicit examples. Furthermore, we obtain explicit parity-dependent closed-form formulas for the reduced first Zagreb index, reduced second Zagreb index, Zagreb coindices and the vertex-PI index of monogenic semigroup graphs. The derivations are based on a neighbourhood-block decomposition of the edge set, which provides a unified computational framework for several families of degree-based topological indices. These results demonstrate how the algebraic structure of a monogenic semigroup determines the behaviour of important graph-theoretical descriptors.

Keywords

• Monogenic semigroups
• Zagreb indices
• PI index

Supporting Institution

This study has not received any financial or institutional support from any organization or project.

Ethical Statement

This study is a purely theoretical mathematics study and does not involve any human or animal subjects, personal data, surveys, interviews, or experimental procedures. Therefore, ethics committee approval is not required.

Thanks

The author declares that there are no specific individuals or institutions to acknowledge for this study.

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