Graphs without dead ends (Q1908274)

The author studies antipodal graphs, i.e. such graphs \(G\) that for any vertex \(u\) of \(G\) there exists a vertex \(\overline u\) of \(G\) such that \(d(u, x)+ d(x, \overline u)= d(u, \overline u)\) for all \(x\in V(G)\). In these graphs, for a given \(u\), there exists exactly one antipode \(\overline u\) with \(d(u, \overline u)= d(G)\). The author characterizes these graphs and discusses properties of the graphs such as degree, girth, and automorphism. He illustrates them with numerous interesting examples.