Journals Information
Mathematics and Statistics Vol. 13(5), pp. 259 - 268
DOI: 10.13189/ms.2025.130501
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Restricted Detour 
-Distance and Restricted Detour -Index of Vertex Identified and Edge Introducing Graphs
Mohammed Rahman Ahmed 1,*, Gashaw Aziz MohammedSaleh 2
1 Department of Mathematics, College of Education, Garmian University, Iraq
2 Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq
ABSTRACT
To establish a mathematical foundation for the extensive research areas of Quantitative Structure鈥揂ctivity Relationship and Quantitative Structure鈥揚roperty Relationship, the concept of chemical graph theory has been introduced. In this framework, topological indices are defined as numerical descriptors that quantitatively represent the structural features of molecular graphs. A wide variety of topological indices have been systematically developed and extensively employed as predictive tools in Quantitative Structure鈥揂ctivity Relationship and Quantitative Structure鈥揚roperty Relationship modeling studies. Given the importance of indices in chemical graph theory, we introduce a novel distance between two vertices 
and 
, which is the restricted detour 
-distance in order to define the restricted detour -index. The restricted detour path 
between two vertices and in a connected graph 
is the longest 
path 
in the graph such that 
. The -distance 
, in which the minimum is taken over all paths and 
is the length of the path . Restricted Detour -distance and Restricted detour -index are two significant concepts in graph theory that provide deeper insights into the structural properties of graphs, particularly in relation to path lengths and network analysis. This paper fills that gap by formulating expressions for the restricted detour -index for two structurally important graph operations: vertex identification and edge introducing. Our results extend known formulas and provide a unified framework for analyzing these operations, which has both theoretical implications and practical applications in network topology and mathematical chemistry. In addition to finding the restricted detour -index of vertex identified and edge introducing graphs, in this article we found the restricted detour -index of some special graphs such as Cycle, Wheel, thorn Path, thorn Rod, Caterpillar graphs and examined related properties.
KEYWORDS
-Distance, Detour -Distance, Restricted Detour -Distance, Restricted Detour -index, Vertex Identified Graph, Edge Introducing Graph
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Mohammed Rahman Ahmed , Gashaw Aziz MohammedSaleh , "Restricted Detour -Distance and Restricted Detour -Index of Vertex Identified and Edge Introducing Graphs," Mathematics and Statistics, Vol. 13, No. 5, pp. 259 - 268, 2025. DOI: 10.13189/ms.2025.130501.
(b). APA Format:
Mohammed Rahman Ahmed , Gashaw Aziz MohammedSaleh (2025). Restricted Detour -Distance and Restricted Detour -Index of Vertex Identified and Edge Introducing Graphs. Mathematics and Statistics, 13(5), 259 - 268. DOI: 10.13189/ms.2025.130501.