Graphical edge-weight-function indices of trees

Abstract

Consider a tree graph $ G $ with edge set $ E(G) $. The notation $ d_G(x) $ represents the degree of vertex $ x $ in $ G $. Let $ \mathfrak{f} $ be a symmetric real-valued function defined on the Cartesian square of the set of all distinct elements of the degree sequence of $ G $. A graphical edge-weight-function index for the graph $ G $, denoted by $ \mathcal{I}_\mathfrak{f}(G) $, is defined as $ \mathcal{I}_\mathfrak{f}(G) = \sum_{st \in E(G)} \mathfrak{f}(d_G(s), d_G(t)) $. This paper establishes the best possible bounds for $ \mathcal{I}_\mathfrak{f}(G) $ in terms of the order of $ G $ and parameter $ \mathfrak{p} $, subject to specific conditions on $ \mathfrak{f} $. Here, $ \mathfrak{p} $ can be one of the following three graph parameters: (ⅰ) matching number, (ⅱ) the count of pendent vertices, and (ⅲ) maximum degree. We also characterize all tree graphs that achieve these bounds. The constraints considered for $ \mathfrak{f} $ are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.

Keywords

• topological index
• graphical edge-weight-function index
• euler-sombor index
• tree graph
• bound