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

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.