JOURNAL OF SCIENCE AND ARTS, no.4, pp.811-822, 2019 (Journal Indexed in ESCI)
Let be a simple graph of order with degree sequence (d) = (d(1), d(2), ..., d(n))) and conjugate degree sequence (d*) (d(1)*, d(2)*, ..., d(n)*). In [1, 2] it was proven that epsilon(G) <= Sigma(n)(i=1) root d(i) and Sigma(n)(i=1) root d(i)* <= LEL(G) <= IE(G) <= Sigma(n)(i=1) root d(i), where epsilon(G) LEL(G) and IE(G) are the energy, the Laplacian-energy-like invariant and the incidence energy of G, respectively, and in [2] 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 epsilon(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 <= n <= 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 >= 1 belong.