A NEW PROOF OF THE SZEGED – WIENER THEOREM

The Wiener index W (G) is the sum of distances between all pairs of vertices of a connected graph G. For an edge e of G, connecting the vertices u and v, the set of vertices lying closer to u than to v is denoted by Ne(u). The Szeged index, Sz(G), is the sum of products |Nu(e)| × |Nv(e)| over all edges of G. A block graph is a graph whose every block is a clique. The Szeged–Wiener theorem states that Sz(G) = W (G) holds if and only if G is a block graph. A new proof of this theorem if offered, by means of which some properties of block graphs could be established.