' -A ON COLOR POLYNOMIALS OF FIBONACCI GRAPHS(U) GEORGIA /1 UNIV ATHENS DEPT OF CHEMISTRY S EL-BASIL it RUG 87 TR-52 N K-8365

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1 ' -A ON COLOR POLYNOMIALS OF FIBONACCI GRAPHS(U) GEORGIA /1 UNIV ATHENS DEPT OF CHEMISTRY S EL-BASIL it RUG 87 TR-52 N K-8365 UNCLASSIFIED F/G 12/1 NL l,

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3 iiic FILE CO OFFICE OF NAVAL RESEARCH Contract N K-0365 R&T Code Technical Report No. 52 On Color Polynomials of Fibonacci Graphs (V' by DTIC Sherif El-Basil S ELECTE 0 'V Prepared for Publication ,.in the o Jounal of Computational Chemistry %I University of Georgia Department of Chemistry Athens, Georgia August 11, 1987 Reproduction in whole or in part is permitted for any purpose of the United States Government This document has been approvee for public release and sale; its distribution is unlimited. IIIs

4 SECulRI T v C, Unclassified ASSIFICATION. I V TI1S PAGE f"l9n Does Entered) REPOT DCUMNTATON AGEREAD REPOT DCUMNTATON AGEBEFORE INSTRUCTIONS COMPLETING FORM I. REPORT Nu aer 2.0VACS0 3. RECIPIIENT'S CATALOG NUMBIER Technical Report No. 52 A KA c,& _/_ 4. TITLE (aend Subtitle) W,,S TYP E OF REPORT II PERIOD COVERED On Color Polynomials of Fibonacci Graphs Technical Report 6. PERFORMING ORO. REPORT NUMBER 7, AUTmOal/s) 8. CONTRACT OR GRANT NUMIER(S) Sherif El-Basil N K PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK AREA & WORK UNIT NUMBERS University of Georgia Department of Chemistry Athens, GA CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Office of Naval Research August 11, 1987 Department of the Navy 13 NUMBEROFPAGES Arlington. VA MONIT 6RING AGENCY NAME & ADDRESS(I1 different from Controlling Office) 1S. SECURITY CLASS. (of this report) ISa DECL ASSIFICATION, DOWNGRADING SCHEDULE 16 DISTRIBUT ON STATEMENT (of this Report) This document has been approved for public release and sale; its distribution is unlimited. 17 DISTRIBUION STATEMENT (of the abstract entered In Block 20, If different from Report) TO SUPPLEMENTARY NOTES To be published in Journal of Computational Chemistry 1S. KEY WORDS (Continue on reverse side If necessary and identity by block number) Graph Theory Fibonacci Graphs Color Polynomials King Polyomino Graphs 20 ABSTRACT (Continue on reverse side It necessary and Identity by block number) A recursion exists among the coefficients of the color polynomials of some of the families of graphs considered in recent work of Balasubramanian and Ramaraj 1. Such families of graphs have been called Fibonacci graphs. Application to king patterns of lattices is given. The method described here applies only to the so called Fibonacci graphs. DO ',AAR", EOITION O 1 NOV S N LF S OBSOLETE Unclassified SECURITY CLASSIFICATION OF THIS PAGE (When Data Inltwd)

5 On Color Polynomials of Fibonacci Graphs Sherif El-Basil* Chemistry Department, University of Georgia Athens, GA U.S.A. Abstract A recursion exists among the coefficients of the color polynomials of some of the families of graphs considered in recent work of Balasubramanian and Ramaraj I. Such families of graphs have been called Fibonacci graphs. Application to king patterns of lattices is given. The method described here applies only to the so called Fibonacci graphs. F fntis CR, &I S1 IC TA3 '4' *t A : Fay of P y, 'I '" N *PraetAdes aclyo hray as IAn t r, Egp - ~~Z k~k~:x~ ~K~

6 -2-1. Introdution Recently Balasubramanian and Ramaraj 1 wrote an Interesting paper on a newly defined color polynomial of certain graphs. They related their work to the pioneering work of Motoyama and Hosoya 2 on king polynomials. Their paper has its merits in both the areas of statistical mechanics and "chemical' graph theory. The purpose of this communication is to cite an observation on a recursive relation occurring amont the coefficients of the color polynomials of some of the families of graphs and their corresponding king patterns which they considered. The observation may be of value from both the computational and graph-theoretical viewpoints. The method which will be described here applies only to the so called Fibonacci graphs. 3.' 2. Definition of Fibonacci Graphs 3 In a homologous series of graphs the set {Gn, Gn+ 1, Gn+2,... } where the number of vertices, n, may or may not be finite, has been called a set of Fibonacci _ail2jh 3 if the following recursion is satisfied: e (Gn+2, k+1) = e(gn+l, k+) + 6 (Gn,k) (1) where e(g,k) is some graph-theoretical invariant of G following: which may include the i) The number of k-matchings 4 in a graph ii) The number of k mutually resonant but nonadjacent sextets when G=B, a benzenoid system iii) The number of k independent sets of vertices when G=C, the so called Clar graph 5 ' 6. A.. %

7 -3- Inter-relations among these Invariants have been recently published 7. Hosoya 8 seems to be the first who observed recursive relations of the type of eqn. 1 but only for the paths and the cycles when e(gk) becomes the number of matchings and G is either a path or a cycle. Recently this author 3 and Gutman 9 generlized the concept to other types of graphs which obey eqn. (1) and to several graph invariants. 3. Construction of Fibonacci Graphs The (finite or infinite) set {Gn, Gn+ 1,... Gn+S}, n > 0, s > n+1 is called a set of Fibonacci graphs. Further, if either vo or v 1 is of degree one, then also { G- 1,..., G n ) is a set of Fibonacci graphs. Such a set must possess at least three elements. The above construction is illustrated in Fig. 1 on the molecular graph of the benzyl radical. There are two modes of graph growth leading to Fibonacci graphs, i.e. "Fibonacci growth", viz., (a) external graph growth (path * : growth) and (b) internal graph growth (cycle growth). 4. Application to Color Polynomials 1 and king Patterns1,2 First we observe that the color polynomials given in ref. 1 are equivalent to the independence polynomials 5, 6 introduced earlier. Thus O(G,k) is defined 5,6 'I" to be the number of selections of k independent vertices from G. This is preciely the number of ways of coloring k vertices black so that no two black vertices are adjacent. Table V11 of ref. 1 lists color polynomials of some cycles. Of course a homologous series of rings form a set of Fibonacci graphs and thus should %conform to eqn. 1 where l(g,k) = 0(C;k), C = c-cle* The coefficients (i.e. O(C;k)'s) *Balasubramanian and Ramaraj 1 have shown that the coefficients of the color polynomials of the paths are the Fibonacci numbers while those of the cycles are manage numbers.

8 -4- are reproduced here to demonstrate the validity of eqn. 1. il _ C 20 0 As a further application of the concept of Fibonacci graphs we calculate the color polynomial of G ; a graph on 25 vertices. 'l,14 There are a number of routes for the homolgation to G 10,14 from smaller graphs. One such route is indicated below S12 Inle-ol - Growths in ring A -G C,1 4 G2 1 4 G o, 14 Honologation to G2,14 is shown in Table 1. To obtain G10,14 from G2,14 we need the color polynomial of G3,14 which is calculated using recursion 2 7

9 C(G;x) = C(G-v;x) + xc(ge v;v) (2) where C(G;x) is the cycle polynomiall, 6, 7 of Gand other symbols have their usual meanings. If one chooses the tetravalent vertex the polynomial Is obtained in terms of (the known) path polynomials: C(G 3, 14 ;x) = X + 134X X X X X X X 8 + 8x 9 Then G2,14 and G2,15 are the first two leading Fibonacci graphs for the second p' Internal growth in ring B (Table 2). Obviously G10,14 corresponds to the lattice in Fig. 2. p. 5. Conclusion Recursive relations of form 1 are very helpful in construction of counting polynomials of potentially very large graphs. Such a buildup from very small units is nceptually similar to expanding the secular determinant of a graph by pruning it down to smaller fragments 10. The identification of a particular family of a Fibonacci graph is certainly of topological and computational importance and is probably equivalent to a botanical identification of a plant family. Acknowledgments I thank the U.S. Office of Naval Research for partial support of this work. '" Illuminating discussions of Professor R.B. King are appreciated. Travel assistance from Fullbright Commission in Cairo is acknowledged. i4*

10 -6-1. K. Balasubramanlan and R. Ramaraj, J. Comput. Chem..L 447 (1985). 2. A. Motoyama and H. Hosoya, J. Math. Phys..L 1485 (1985). 3. S. El-Basil, Theoret. Chim. ActalL. 191 (1984),_EL 199 (1984). 4. H. Hosoya, Bull. Chem. Soc. Japan, AA (1971) Gutman, Z. NaturforschlZjL 69 (1982) Gutman and S. El-Basil, Z. Naturforsch Iii. 276 (1984). 7. S. El-Basil, J. Chem. Soc., Faraday Trans. 2, (1986). 8. H. Hosoya, The Fibonacci Quarterly, ia, 173 (1976) Gutman and S. El-Basil, Math. C hem. in press. 10. K. Balasubramanian and M. Randi4 Theoret. C him. Acta,iL 307 (1982). 'Ah..K ja.

11 The two types of Fibonacci growths of graphs: (a) External subdivision and (b) Internal subdivision. Fig. The lattice graph corresponding to G 10,1 4. There are king patterns generated when 6 kings assume nontaking positions. (c.f. Tables I and 2). Observe that G10,14 is the dualist graph of the above lattice. 'a. r I a.' V am 1 V

12 Homologation from G2,2 to G2,14. Numbers are coefficients of color polynomials. Relation 1 is observed throughout. The computation involves 12 "Fibonacci-growths" "' A. I,,

13 Homolgation G 2, G 1 0,1 4 via 8 internal Fibonacci growths. Numbers are coefficients of color polynomials w I - II , 25, 274, 1732, 6989, 18822, 34362, 42344, 34438, 17689, 5320, 819, 48, 0. -A..,',,, Np. J..

14 CCD) CC G*C) 00 C )

15 'p -4-. ", (4.- -

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