Generalized graphs and the Sinanoğlu graphical rules
Mingzuo Shen · International Journal of Quantum Chemistry · 1990
Abstract A comparison of Sinanoğlu's VIF (Ref. 1) and generalized graph is presented. Generalized graphs have vertex and edge weights. An abridged history of generalized graphs in theoretical chemistry is given. VIF's are generalized graphs and therefore have adjacency matrices. The “graphical” rules of Sinanoǧlu can be represented by congruent transformations on the adjacency matrix. Thus the method of Sinanoǧlu is incorporated into the broad scheme of graph spectral theory. If the signature of a graph is defined as the collection of the number of positive, zero, and negative eigenvalues of the graph's adjacency matrix, then it is identical to the all‐important {n+, n0, n−}, the {number of positive, zero, and negative loops of a reduced graph} or the {number of bonding, nonbonding, and antibonding MOs}. A special case of the Sinanoğlu rules is the “multiplication of a vertex” by (−1). In matrix language, this multiplication is an orthogonal transformation of the adjacency matrix. Thus, one can multiply any vertex of a generalized graph by −1 without changing its eigenvalues.