LOGARITHMIC ASYMPTOTICS OF THE NUMBER OF CENTRAL VERTICES OF ALMOST ALL n-VERTEX GRAPHS OF DIAMETER k

The asymptotic behavior of the number of central vertices and F. Buckley's central ratio R-c(G)= vertical bar C(G)vertical bar/vertical bar V(G)vertical bar for almost all n vertex graphs G of fixed diameter k is investigated. The logarithmic asymptotics of the number of central vertices for almost all such n-vertex graphs is established: 0 or log(2) n (1 or log(2) n), respectively, for arising here subclasses of graphs of the even (odd) diameter. It is proved that for almost all n-vertex graphs of diameter k, R-c(G) = 1 for k = 1, 2, and R-c(G) = 1 - 2/n for graphs of diameter k = 3, while for k >= 4 the value of the central ratio R-c(G) is bounded by the interval (Delta/6 + r(1) (n), 1 - Delta/6 - r(1) (n)) except no more than one value (two values) outside the interval for even diameter k (for odd diameter k) depending on k. Here Delta is an element of (0, 1) is arbitrary predetermined constant and r(1) (n), r(2) (n) are positive infinitesimal functions.