Topological indices are numeric values associated with a graph and characterize its structure. There are various topological indices in graph theory such as degree-based, distance-based, and counting-related topological indices. Among these indices, degree-based indices are very interesting and studied well in literature. In this work, we studied the generalized form of harmonic, geometric-arithmetic, Kulli–Basava indices, and generalized power-sum-connectivity index for special graph that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. We found exact values for the stated indices and for the stated special graphs. We also investigated the generalized form of the indices for various properties of alkane isomers, from which we obtained interesting results which are closed to that of experimental obtained results.

Interconnection systems in computer science and information technology are mainly represented by graphs. One such instance is of swapped network simulated by the optical transpose interconnection system (OTIS). Fault tolerance has become a vital feature of optoelectronic systems. Among multiple types of faults that may take place in an interconnection system, two significant kinds are either due to malfunctioning of a node (processor in case of O G ) or collapse of communication between nodes (failure of interprocessor transmission). To prevail over these faults, the unique recognition of every node is essential. In graph-theoretic interpretation, this leads to instigating the metric dimension β O G and fault-metric dimension β   ′ O G of the graph O G obtained from the interconnection system. This paper explores OTIS over base graph P m (path graph over m vertices) for resolvability and fault-tolerant resolvability. Furthermore, bounds for β O G and β ′ O G are also imparted over G = P m .

Molecular descriptors are a basic tool in the spectral graph, molecular chemistry, and various other fields of mathematics and chemistry. Kulli–Basava K B indices were initiated for chemical applications of various substances in chemistry. For simple graph G , K B indices in generalized forms are K B 1 ϱ G = ∑ g h ∈ E G S e g + S e h ϱ and K B 2 ϱ G = ∑ g h ∈ E G S e g . S e h ϱ , where S e g = ∑ e ∈ N e g d G e , and for edge e = g , h , the degree is d G e = d G g + d G h − 2 and ϱ ≠ 0 is any real number. The graph G is said to be a k − g e n e r a l i z e d quasi tree if for the vertex set U k ⊂ G having U k = k , G − U k is a tree and for U k − 1 ⊂ V G having U k − 1 = k − 1 , G − U k − 1 is not a tree. In this research work, we have successfully investigated sharp bounds of generalized K B indices for k-generalized quasi trees where ϱ ≥ 1 . Chemical applications of the generalized form are also studied for alkane isomers with scatter diagrams and residuals.