The degree distance index (DDI) is a vertex-degree weighted version of a well-known index that is called by Wiener index (WI). In extremal theory of graphs, improving the bounds with best possible values is a worth investigating problem. In this note, the exact formulae of the DDI for the four different types of the sum graphs in the terms of various indices of their factor graphs are computed. Moreover, a comparison is also presented between the obtained exact and already existing bounded values for the particular sum graphs.
In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve the problems existing in various disciplines of physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one of the graph invariants which are frequently used to identify the different physicochemical and structural properties of molecular graphs. Wiener index is the first distance-based TI that is used to compute the boiling points of the paraffine. For a graph F, the recently developed Gutman Connection (GC) index is defined on all the unordered pairs of vertices as the sum of the multiplications of the connection numbers and the distance between them. In this note, the GC index of the operation-based symmetric networks called by first derived graph D1(F) (subdivision graph), second derived graph D2(F) (vertex-semitotal graph), third derived graph D3(F) (edge-semitotal graph) and fourth derived graph D4(F) (total graph) are computed in their general expressions consisting of various TIs of the parent graph F, where these operation-based symmetric graphs are obtained by applying the operations of subdivision, vertex semitotal, edge semitotal and the total on the graph F respectively.
The idea of soft set was initiated by Molodstov. Soft sets have been used for decision making in dealing vague ideas. In this paper, soft sets are represented by bipartite graph. Operations on soft sets are also represented by bipartite graph.
Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.
Distance based topological indices (TIs) play a vital role in the study of various structural and chemical aspects for the molecular graphs. The first distance-based TI is used to find the boiling point of paraffin. The connection distance (CD) index is a latest developed TI that is defined as the sum of all the products of distances between pair of vertices with the sum of their respective connection numbers . In this paper, we computed CD indices of the different derived graphs (subdivision graph S G , vertex-semitotal graph R G , edge-semitotal graph Q G and total graph T G obtained from the graph G under various operations of subdivision in the form of degree distance (DD) and CD indices of the basic graphs including some other algebraic expressions.
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