DISTANCES AND VOLUMINA FOR GRAPHS

It has long been realized that connected graphs have some sort of geom etric structure, in that there is a natural distance function (or metr ic), namely, the shortest-path distance function. In fact, there are s everal other natural yet intrinsic distance functions, including: the resistance distance, correspondent ''square-rooted'' distance function s, and a so-called ''quasi-Euclidean'' distance function. Some of thes e distance functions are introduced here, and some are noted not only to satisfy the usual triangle inequality but also other relations such as the ''tetrahedron inequality''. Granted some (intrinsic) distance function, there are different consequent graph-invariants. Here attent ion is directed to a sequence of graph invariants which may be interpr eted as: the sum of a power of the distances between pairs of vertices of G, the sum of a power of the ''areas'' between triples of vertices of G, the sum of a power of the ''volumes'' between quartets of verti ces of G, etc. The Cayley-Menger formula for n-volumes in Euclidean sp ace is taken as the defining relation for so-called ''n-volumina'' in terms of graph distances, and several theorems are here established fo r the volumina-sum invariants (when the mentioned power is 2).