Extremal Problems for Graphical Function-Indices and f-Weighted Adjacency Matrix

Abstract

Abstract Let f(x, y) (f(x)) be a symmetric real function (real function) and G = (V,E) be a graph. Denote by di the degree of a vertex i in G. The graphical function-index TIf (G) (Hf (G)) of G with edge-weight (vertex-weight) function f(x, y) (f(x)) is defined as TIf (G) = ∑ uv∈E f(du, dv) (Hf (G) = ∑ u∈V f(du)). We can also get a weighted adjacency matrix from the edge-weighted graph, i.e., Af (G) = (afij) where a f ij = f(di, dj) if vertices i and j are adjacent in G, and 0 otherwise. This matrix is simply referred to as the f -weighted adjacency matrix. One can see that the concepts of graphical function-indices and f -weighted adjacency matrix can cover all the degree-based graphical indices and degree-based adjacency matrices of graphs, such as the Zagreb indices, Randić index, ABC-index, etc., and the Randić matrix, ABC-matrix, GA-matrix, etc. So, for the graphical function-indices TIf (G) and Hf (G) and the f -weighted adjacency matrix Af (G) of a graph G, one can think about finding unified ways to study the extremal problems and spectral problems. This survey is intended to sum up the results done so far on these problems.