The Wiener index is a graph invariant that has found extensive application in chemistry. In addition to that a generating function, which was called the Wiener polynomial, who’s derivate is a q-analog of the Wiener index was defined. In an article, Sagan, Yeh and Zhang in [The Wiener Polynomial of a graph, Int. J. Quantun Chem., 60 (1996), 959969] attained what graph operations do to the Wiene...

We introduce a notion of compatibility for multiplicity matrices. This gives rise to necessary condition the join two (possibly disconnected) graphs G and H be pattern an orthogonal symmetric matrix, or equivalently, minimum number distinct eigenvalues q G∨H equal two. Under additional hypotheses, we show that this is also sufficient. As application, prove q(G∨H) either three when are unions co...