Stolarsky-3 mean status connectivity index of graphs with the application on alkanes of <i>C</i> _<i>n</i> <i>H</i> _2<i>n</i>+2

Abstract Topoligical indices play a crucial role in chemical graph theory, aiding in the characterization of molecular structures and the prediction of their physicochemical properties. This paper introduces the Stolarsky-3 mean status connectivity index, denoted as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>S</mml:mi> <mml:mi>M</mml:mi> <mml:msub> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mo>∑</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>u</mml:mi> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:msqrt> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>σ</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>σ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msqrt> </mml:math> , a novel class of distance-based descriptors that incorporate the Stolarsky mean with the status of a vertex. The Stolarsky mean is a powerful mathematical function that generalizes several classical means, allowing a smooth transition between different types of averaging methods. By incorporating this mean with vertex status, the proposed index enhance the ability to distinguish molecular structure more effectively than traditional connectivity indices. The status of a vertex is defind as the sum of its distances to all other vertices in the graph, reflecting its relative centrality or importance within the molecular structure. In this work, We also establish the bounds for the proposed index, compare it with traditional indices, and determine the Stolarsky-3 mean status connectivity index for several standard graphs. Specifically, the status and Stolarsky-3 mean status connectivity index are computed for caterpillar gtraphs. Using these computations, We establish a regression model relating Stolarsky-3 mean status connectivity index to the physical properties of alkanes of C n H 2 n +2 . Furthermore, We compare the SMS 3 index with various types of topological indices for the OTIS swapped network using a path-based approach.