B) a graph we mean a finite undirected connected graph of order p, p 2 2, with no loops or multrple edges. A finite non-decreasing sequence S : s,. s:.. . , sp p * 2. of positive integers is an eccentric sequence if there exists a graph G with vertex set V(G) = {ul, o_, . . . . u,,} such that for each i 1 s i d p, s, IS the e+:centricity of 0,. A set S is an eccentric set if there exists a graph G such that the eccentricity e( u, ) is in S for every ~1, E V(G), and even element of S is the cccentrrcity of *3rne vcrte-t in G. In this note we characterize eccentric sets. and we find the mnnmum order among all graphs whose eccentric set is a given set, to obtain a new necessary condition for a sequence to he eccentric. We also present some properties of graphs having prea\srgncd eccentrrc xquences.
I. lntmduction
Let t'E V(G). Then o(
c)= max(d(u, v): u E V(G)), where d(u, v) is the distance from u to 1~. is the eccenm'city of the vertex v. A vertex of G is a cenlnal vertex if it has minimum eccentricity in G. The set of central vertices of G is the cenfer of G. An analog of the degree sequence of a graph [3, 51 is its eccvnfric sequenctp S(G): e(vI), e(v.!), . . . , e@,,). where V(G) = {v!,. . . ,.vJ. From here on, by a sequence we will always mean a finite non-decreasing sequence of positive integers A sequence S: s!. So,. . . , s, p 2 2, is an eccentric sequence if there exists a graph G whose vertices (;an be lahelled vl, vz, . . . , v, so that e( v, ) = s, for each i, I G i s c. Among other results the following three necessary conditions for a sequence to be eccentric are presented in (7): (i) c' * 2s,, (ii) s, s min(p -I, 2sJ, (iii) for everv integer k. sI < k s s,, there exists some i, 2 s i s p -1, such that F . # =s *,*l = k. ' An analog of the notion of degree set for graphs and digraphs [2,6] is the corlcept of eccentric set. A finite non-empty set S of positive integers (no repetitions) is an