The Topological Indices Of Signed Graphs

Abstract

The topological indices are the numbers generated from several invariants of molecular graphs that represents important information about the chemical structures or molecules. The first topological index was recognized in chemical graph theory in 1947 by Harry Wiener. The name signed graph and the notion appeared first in a mathematical paper of Frank Harary in 1953. A signed graph is a graph in which each edge has a positive or negative sign. In our thesis, we introduce the vertex labeled signed graphs, vertex labelled signed fuzzy graphs and vertex labeled signed domination graphs for the topological indices by comparing them with some standard graphs and chemical graphs. newlineIn Chapter 1, not only we will present the necessary terminology and notations that are not typically found in normal textbooks of Graph Theory, but we will also discuss fundamental or well-known concepts and conclusions that will be referred to in following chapters of the thesis. newlineIn Chapter 2, we define first modified Zagreb index, second modified Zagreb index, first Zagreb complexity index and second Zagreb complexity index for degree based vertex labeled graphs. We also compute degree based vertex labeled Zagreb indices for some standard graphs and some chemical signed graphs. newlineIn Chapter 3, we study connectivity indices for degree based labeled signed graphs. We also compute various results for these degree based labeled topological indices for some standard signed graphs and some chemical signed graphs as discussed in Chapter 2. newlineIn Chapter 4, using the idea of signed degree, we define the first and second Zagreb indices, the Randic connectivity index, the Sum-connectivity index, the Harmonic index, and the Weiner index for fuzzy labeled signed graphs and some chemical graphs. newlineIn addition to calculating the molecular descriptors, we provide the procedure for determining them. Several generalizations are made between all these indices.