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Generalized graphs and the Sinanoğlu graphical rules
Author(s) -
Shen Mingzuo
Publication year - 1990
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560370204
Subject(s) - adjacency matrix , graph energy , combinatorics , mathematics , regular graph , eigenvalues and eigenvectors , adjacency list , vertex (graph theory) , two graph , discrete mathematics , graph , line graph , voltage graph , graph power , physics , quantum mechanics
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 + , n 0 , n − }, the {number of positive, zero, and negative loops of a reduced graph} or the {number of bonding, nonbonding, and antibonding MO s}. 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.