On the eccentricity matrix of graphs and its applications to the boiling point of hydrocarbons

The eccentricity matrix E(G) of a graph G is derived from the distance matrix by keeping for each row and each column only the largest distances and leaving zeros in the remaining ones. The E-eigenvalues of a graph G are those of its eccentricity matrix E(G). The E-spectrum of the graph G is the multiset of its E-eigenvalues, where the maximum modulus is called the E-spectral radius of G. In this paper, we characterize the graphs whose E-spectral radius attains the minimum (resp. the second minimum) value and the graphs maximizing the least and the second least E-eigenvalues. As a by-product, the graphs with xi(G) = diam(G) or zeta(G) =-diam(G) for diam(G) is an element of {1, 2} are identified, where diam(G) denotes the diameter of G. Furthermore, we determine the graphs other than K-1,K-n1,K-...,K-n6 (n(i) >= 2, i is an element of {1, ..., 6}) whose least E-eigenvalue is equal to -2 root 2 . Additionally, the E-spectral determination of graphs is investigated. At last some numerical results are discussed, in which the linear models for the E-spectral radius (resp. E-energy) are better than or as good as the models corresponding to the spectral radius (resp. energy) in terms of some parameters.