SOME RESULTS AND A CONJECTURE ON CERTAIN SUBCLASSES OF GRAPHS ACCORDING TO THE RELATIONS AMONG CERTAIN ENERGIES, DEGREES AND CONJUGATE DEGREES OF GRAPHS


ALTINIŞIK E., Varlioglu N. M.

JOURNAL OF SCIENCE AND ARTS, no.4, pp.811-822, 2019 (Peer-Reviewed Journal) identifier

  • Publication Type: Article / Article
  • Volume: Issue: 4
  • Publication Date: 2019
  • Journal Name: JOURNAL OF SCIENCE AND ARTS
  • Journal Indexes: Emerging Sources Citation Index
  • Page Numbers: pp.811-822

Abstract

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.