Research On The Extremal Problems Of Two Types Of Distance-based Graph Parameters

In 2010, Hansen et al. proposed three conjectures on the differences between the (revised) Szeged index and the Wiener index for a connected graph G. Recently, the above conjectures were solved by Chen et al. in [L.L. Chen, X.L. Li, M.M. Liu, The (revised) Szeged index and the Wiener index of a non-bipartite graph, European J. Combin.36 (2014) 237-246]. In this paper, as a continuance of it, we study some further relation between the (revised) Szeged index and Wiener index of connected graphs. Some sharp bounds on the difference between the (revised) Szeged index and Wiener index are established and the corresponding extremal graphs are characterized.The total reciprocal edge-eccentricity is a novel graph invariant with vast poten-tial in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical proper-ties. In this paper we first introduced four edge-grafting transformations to study the mathematical properties of the reciprocal edge-eccentricity of G. Using these nice mathematical properties, we characterize the extremal graphs among n-vertex trees with given graphic parameters, such as pendants, matching number, domina-tion number, diameter, vertex bipartition, et al. Some sharp bounds on the reciprocal edge-eccentricity of trees are determined. The concrete content is in the following:● In Chapter 1, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discussion, by using deep-going analysis, it fully shows the main work’s necessity and innovation.● In Chapter 2, we give some necessary definition and lemmas.● In Chapter 3, we introduce the bounds on the difference between the (re-vised)Szeged index and Wiener index of graphs and identify the responding extremal graphs.● In Chapter 4, we introduce four edge-grafting transformations to study the mathematical properties of the total reciprocal edge-eccentricity of graphs.● In Chapter 5, sharp bound is established on the total reciprocal edge-eccentricity of n vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. The corre-sponding extremal graphs are identified, respectively.●In Chapter 6, we summarize the main results in this paper and give some prospects for further research in the future.