Realizing the distance matrix of a graph

An explicit description is giv e n for th e uniqu e gra ph with as few arcs (eac h bearin g a positive length) as pos s ibl e, whi c h has a presc rib ed mat rix of s hortest-p ath di stan ces be twee n pa irs of distinct vertices. The sam e is d one in th e case wh e n the ith diago na l matrix e ntr y, in s te ad o f be ing zero , represents th e. le ngth of a s hort est c losed path co ntainin g th e ith vertex. Ke y Word s: Graph, di s ta nce ma trix , s hortes t path. Le t G be a finite oriented graph with verti ces {Vi}~', wh e re n > 2. To avoid unn ecessary co mpli cation s, we res tric t attention to connected graph s, i. e., if i r!= j then G co ntain s a directed path from Vi to Vj. As add iti onal s tru cture, we assume associated to G a positive-valu ed fun cti on lc ass ignin g lengths lc(i, j) to the arcs (Vi, Vj) of G. The distance matrix Dc of G has e ntri es dc;(i , i) = ° on th e main diago nal; a typi c al off-diago nal e ntry dc(i, J) re pers e nts the le ngth of a s hortes t directed path in G from Vi to Vj. An arc of G is called redundant if its deletion leaves Dc un changed. Th e graph G will be called irreducible if it co ntain s no redundant arcs. A real square matrix D with e ntri es d(i , j) is called realizable if there is a grap h G s uc h that D = Dr;. Hakimi and Yau t showed that necessary and s uffi cie nt conditions for th e realiza bility of Dare The necessity of the se conditions should be clear. To prove sufficiency one need only take the arcs of G to be all (Vi. Vj) with i r!= j , and define le by le/i, J) = d(i , j) ; it follows readily from (1) to (3) that Dc= D. If matrix D is realizable, it clearly has a realization by …