Some results and a conjecture on certain subclasses of graphs according to the relations among certain energies, degrees and conjugate degrees of graphs

TL;DR

This paper investigates subclasses of simple graphs based on relations among energies, degrees, and conjugate degrees, providing counts for small graphs, conjectures on asymptotic ratios, and classifying specific graph types.

Contribution

It characterizes subclasses of graphs according to energy relations, counts their occurrences for small n, and proposes a conjecture on their asymptotic distribution.

Findings

Counted graphs in each subclass for n up to 8

Conjectured ratios of subclasses as n approaches infinity

Classified paths, cycles, and complete graphs into subclasses

Abstract

Let $G$ be a simple graph of order $n$ with degree sequence $(d)=(d_1,d_2,\ldots,d_n)$ and conjugate degree sequence $(d^*)=(d_1^*,d_2^*,\ldots,d_n^*)$. In \cite{AkbariGhorbaniKoolenObudi2010,DasMojallalGutman2017} it was proven that $\mathcal{E}(G)\leq \sum_{i=1}^{n} \sqrt{d_i}$ and $\sum_{i=1}^{n} \sqrt{d_i^*} \leq LEL(G) \leq IE(G) \leq \sum_{i=1}^{n} \sqrt{d_i}$, where $\mathcal{E}(G)$, $LEL(G)$ and $IE(G)$ are the energy, the Laplacian-energy-like invariant and the incidence energy of $G$, respectively, and in \cite{DasMojallalGutman2017} it was concluded that the class of all connected simple graphs of order $n$ can be dividend into four subclasses according to the position of $\mathcal{E}(G)$ in the order relations above. Then, they proposed a problem about characterizing all graphs in each subclass. In this paper, we attack this problem. First, we count the number of graphs of order $n$ in each of four subclasses for every $1\leq n \leq 8$ using a Sage code. Second, we present a conjecture on the ratio of the number of graphs in each subclass to the number of all graphs of order $n$ as $n$ approaches the infinity. Finally, as a first partial solution to the problem, we determine subclasses to which a path, a complete graph and a cycle graph of order $n\geq 1$ belong.