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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 andthe incidence energy of $G$, respectively, and in \cite{DasMojallalGutman2017}it was concluded that the class of all connected simple graphs of order $n$ canbe dividend into four subclasses according to the position of $\mathcal{E}(G)$in the order relations above. Then, they proposed a problem aboutcharacterizing all graphs in each subclass. In this paper, we attack thisproblem. First, we count the number of graphs of order $n$ in each of foursubclasses for every $1\leq n \leq 8$ using a Sage code. Second, we present aconjecture on the ratio of the number of graphs in each subclass to the numberof all graphs of order $n$ as $n$ approaches the infinity. Finally, as a firstpartial solution to the problem, we determine subclasses to which a path, acomplete graph and a cycle graph of order $n\geq 1$ belong.

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