Theory and computational applications of Fibonacci graphs

The concept of Fibonacci graphs introduced and developed by this author is critically reviewed. The concept has been shown to provide an easypencil-and-paper method of calculating characteristic, matching, counting, sextet, rook, color and king polynomials of graphs of quite large size with limited connectivities. For example, the coefficients of the matching polynomial of 18-annuleno—18-annulene can be obtainedby hand using the definition of Fibonacci graphs. They are (in absolute magnitudes): 1, 35, 557, 5337, 34 361, 157 081, 525 296, 1304 426, 2 416 571, 3 327 037, 3 362 528, 2 440 842, 1 229 614, 407 814, 81936, 8652, 361, 3. The theory of Fibonacci graphs is reviewed in an easy and detailed language. The theory leads to modulation of the polynomial of a graph with the polynomial of a path.Dedicated to Professor R. Bruce King for his enthusiastic promotion and contributions to Mathematical Chemistry.