Extremal graphs for vertex-degree-based invariants with given degree sequences

TL;DR

This paper investigates extremal graphs with given degree sequences that maximize or minimize a vertex-degree-based connectivity function, providing theoretical results and applications for specific graph classes.

Contribution

It establishes the existence of BFS-graphs that extremize the connectivity function for certain degree sequences and proves majorization theorems for unicyclic and bicyclic graphs.

Findings

Existence of BFS-graphs with extremal connectivity functions.

Majorization theorems for unicyclic and bicyclic degree sequences.

Applications to degree-based graph invariants.

Abstract

For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $\pi=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invariants based on degree are included.