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Spectral moments and walks for large carbon cage clusters
Volume 175, number 3
CHEMICAL PHYSICS LETTERS
7 December 1990
Spectral moments and walks for large carbon cage clusters K. Balasubramanian
’
Department of Chemistry, Arizona State University, Tempe, AZ 85287-1604, USA Received 1 August 1990; in final form 6 September 1990
Spectral moments and walks for two forms of the Cso cluster (football form and hand form) as well as a form of ClzOcluster are obtained. It is shown that spectral moments and other topological characteristics of these clusters can be readily obtained using vector&d computer codes.
1. Introduction Ever since Smalley and coworkers [ l-31 have generated the unusually stable Cho cluster buckminsterfullerene (footballene), there have been numerous experimental and theoretical studies on large carbon cage clusters [4-141. In addition to the aesthetic appeal of the football molecule, the generation of this molecule has particularly stimulated graph theoretical and other topological studies in the characterization of large cluster graphs of icosahedral and dodecahedral symmetries. The spectral moments of a graph are fundamental structural invariants (although not unique) and are not only useful in the topological characterization of the connectivity networks hidden in the structure but also in numerous other applications ranging from enumeration of self-returning walks (the famous P6lya problem ) , rapid computation of energy eigenvalues through spectra moments to molecular coding, etc. [ 15-261. Burdett and Lee [ 241 have shown that important insights into similarity of energy difference curves of molecules can be obtained through the various spectral moments of the associated molecular graphs. The set of all spectral moments of a molecular structure can be utilized to construct the electronic density of states. If two graphs have the same set of spectral moments up to mth moment, then it is proposed that only moments of order greater than m will be im’ Camille and Henry Dreyfus Teacher-Scholar.
ponant for the energy difference curves [24]. The energy difference curves obtained from the spectral moments are in turn useful to comprehend the structure of molecules to solid-state materials. The spectral moments of the carbon cluster cages have not been obtained up to now although the eigenvalues and the characteristic polynomials have been obtained before [ 12,131. The objective of this Letter is to show that matrix power methods can be efficiently harnessed to obtain spectral moments and walks on large carbon clusters.
2. Method of enumerating walks and spectral moments A walk on a graph is a sequence of edges a random walker can traverse continuously starting on any vertex and ending on any vertex including the possibility of returning to a vertex visited before during the walk. A self-returning walk is defined as a walk in which the random walker returns to the starting vertex at the end of the walk. The length of a walk is the number of edges traversed in a walk. The kth spectral moment of a graph, S, can be shown to be the number of self-returning walks of length k (for additional discussions on walks and spectral moments see refs. [ 16,17,22]). The walk generating function (GF) Wand the selfreturning walk generating function SRW can be defined as
0009-2614/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
273
Volume 175, number 3
CHEMICAL PHYSICS LETTERS
5 Nkxk,
W=
k-0
SRW= i &Xk, k=O
where Nk is the number of possible walks of length k in the given graph and S, is the number of self-returning walks (spectra moment) of length k. A vectorizcd computer code to generate the walks and spectral moments of graphs containing 200 or larger number of vertices has been developed [ 27 1. This code is based on the matrix power method. It is based on a theorem [ 28 ] that the sums of powers of the adjacency matrices enumerate walks. In symbols, Nk= C (Ak)ij, ij &= c (Ak),, , i
where Ak is the kth power of the adjacency matrix. There are other techniques available for deriving walks especially ones based on the characteristic polynomials and spectra (see refs. [ 23,241 for details). But the method used here requires no modification for directed and signed graphs. Therefore, computation of various powers of the adjacency matrices of graphs in the above method is required to generate spectral moments and walk GFs. In general the walk GFs can propagate up to infinite order but it suff%zes to stop the evaluation for k= n since.higher walks can be enumerated using the Cayley-Hamilton theorem. Also, we omit the constant term (number of vertices in the graph) in Wand SRW. We also compute spectral moments and walks on the signed form of C6,,_The adjacency matrix of a signed graph is defined as A?)=
0 1 = - 1 = 0
=
introduce directionality to walks thereby yielding effective walks including directions. For several directionally oriented properties such as magnetism, we envisage signed graphs to be useful. A disadvantage with the use of signed graphs is that the spectral moments and walks of signed graphs are dependent on labelling of vertices, in general. For bipartite graphs however, a canonical labelling is possible. For bipartite graphs vertices can be labelled such that odd vertices are connected only to even vertices and vice versa.
3. Results and discussions Fig. 1 shows two forms of Cho clusters called colloquially the foot form and the hand form (see refs. [ 11,121). The foot form is the famous buckminsterfullerene while the hand form is not expected to be as stable due to not only the presence of numerous three-membered rings but also due to the absence of six-membered rings. Fig. 2 shows a structure for the ClzO cluster called the archimedene proposed by Haymet [ 71. We do not expect this form of Clzo to
_-d,
; - _ \.
..;----:
-.I
;’
’ ,
:
,‘: -.-A
?, -r-q*
.:
i_ ‘.
,____; .’ \ I 1
@
. ‘. .
HPNO
Fig.
I. The foot and band forms of C6,,.
ifi=j,
if i # j, i and j are connected and i > j , if i # j, i and j are connected and i c j , otherwise.
The signed graphs appear in many applications including electromagnetic theory and the theory of relativity [ 29 1. Signed graphs characterize additional topological information when ordinary graphs may fail to differentiate these. Furthermore signed graphs 274
7 December 1990
Fig. 2. The archimedene cluster (C120).
Volume
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175, number 3
PHYSICS
be unusually stable due to four-membered rings but it is conceivable that this structure for CLzOis a viable possibility. Tables 1 and 2 show the walk GFs and spectral moments of the ordinary foot and hand forms, respectively of CeO,while table 3 shows the results of the signed form C6,, (foot). We used quadruple arithmetic precision to generate the walks and spectral moments. This is required for these graphs since round off errors tend to propagate. It can be easily shown that for all graphs which contain vertices with the same degree (valence) d, the walk GF is given by v31
540 4860 43740 393660 3542940 31886460 286978140 2582803260 23245229340 209207064060
1990
Table 2 The first 58 spectral moments of the hand form of C, (fig. 1)
Table 1 Walks and spectral moments of the foot form of Cso first 20 coefficients in the walk GF 180 1620 14580 131220 1180980 10628820 95659380 860934420 7748409780 69735688020
7 December
LETTERS
0
180
120 1200 10080 80760
900 5340 35460 253620
640200 5093400 40898880
1905660 14811060 117929220
332022240 2724768720 22584987720 188865450360
956060220 7859799300 65340031860 548164718940
1591637358600 13503869994000 115244202689040 988576437445440
4633975769460 3942832 1305860 337353046060380 2900496236873220
8518588872039960 73700566087539240 639933146362108920
25044939940062900 217080643840389180 1888005056872838580
5574450244132047840 48701071093645377600 426606137649840554160
144093969478771638780
3745976162778151137000 32965588082310403244760 290691572663296782772200 2568061851526963237049520 22725431519492831691481200 201414299786386108356241440 787651523135931276819836600
16471000071368648580 1263778538627781666180 11109603786476901689460 97868290409323698899100 863819840896828376802420 7637849892658549126069380 67642704630179032131725340 599947394048498432130601860 5328352422241637700903475380
first 58 spectral moments 0 0 120 1680 18360 184800
180 900 5460 36420 256500 1873620
1783080 16746000 154352520 1403932800 12650599080 113259310320 1009684221720 8977561689600
14073540 108202500 848993460 6783392100 55086863220 453895602420 3788421227940 31979999137860
79711374249240 707384238610800 6278208974261400 55750270236199680
272635097633940 2344134895129860 20302537026863220
495459722454787080 4407471717951260400 39248547319074569640 349878641275493877600 3122194389896999600520 27888692364449629021200 249339995789263026526200 2231091951544049450239200 19978807833978771236840760 179024924705499736515673680 605 15 1473848047475480229240
176934325344415380 1550075679421949700 13639955804155214340 120471316578120932340 1067339198155547291940 9480855692875722930420 84398045708842207276980 752663457823581639132900 6722369998676165376954180 60115767186017862345323220 538156067456971472666165700 4821771501779283290257274100
w=
+&-,
where n is the number of vertices. The expansion of the above function yields W=n( L+du+d2x2+d3x3t...), although in our discussion we omit the constant term. Note that therefore for both the foot and hand forms of CSOthe Ws become identical and the coefficients of xk in both forms are 60 x 3k. The results obtained (only first 20 coefftcients are shown) from our computer codes in table 1 conform to this exact analytical result thereby assuring that both the inputs for these graphs and codes are free of bugs. It is interesting to note that although Ws are identical for the foot and hand form the spectral moments differ except for the first, second and fourth moments of the foot and hand forms. This is comforting since spectral moments have been proposed as molecular descriptors [ 161. We note that the spectral moments of the hand form are in general 215
Volume 175, number 3
CHEMICAL PHYSICS LETTERS
7 December 1990
Table 3 Walks and spectral moments of the signed foot form of C60 (fig. 3 ) first 60 coefficients in the walk GF
first 60 spectral moments
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-76 268 - 1284 6836 -38180 220156 - 1304460 7927372 - 49346244 314190228 -2042844772 13539102332 -91288612844 62498 1158796 -4336268097540 30436050757364 -215760769291300 1542521010952828 -11107139739600332 80463268 154550348 -585865402150167492 4283960873722605524 -31436806791348290916 231378204076767666428 - 1707200681925154806828 12622543907028171672972 -93489115188297469603076 693431455908954915510580 -5149577297325719259529508 38280682790533054885149436
equal or larger than the foot form. Thus there are more possibilities of returning to the starting vertex in the hand form. This is consistent with the connectivity of the hand form as compared to the foot form. That is, the presence of three-membered rings facilitates greater probability for the walker to return to the starting vertex in the hand form compared to the foot form. The walks and spectral moments of the signed form of Ceo with labelling in fig. 3, is shown in table 3. As noted before the walks and spectral moments of signed graphs are label dependent. However, a few terms are independent of the labels. As seen from table 3 all odd walks and spectral moments are zero. This is because the net effect of walking in the same and opposite directions cancel out leading to zero coefficients for all odd number of walks. We tested 276
-180 900 - 5076 30276 - 186900
1182708 - 7632756 50080772 - 333373428 2247905700 -15334130420 105705275700 -735630655316 5163566570628 -36525234178836 260159107586308 -1864491454057556 13435486395672564 -97283418495594164 707395586546432516 - 5162906959438829364 37803236000712441444 -277576997127292038964 2043130837567786737716 -15070410807770717802900 111364305414300031902084 -824234413674118485221460 6IO8651122305790206782404 -45326101689217744335479956 336658469605220442756149748
this out on signed CbOwith different labelling. The coefficients of odd walks were found to be independent of labelling as they vanish. The number of even walks however was found to depend on labelling. This is because for every walk of length k with k, positive steps and k_ negative steps there exists an opposite walk with k_ positive steps and k, negative steps. The effect of the former and latter walks cancel out for walks of odd length. The fifth and sixth spectral moments are the first and second for which the signed and ordinary forms of CGOdiffer. A similar behavior was noticed for the hand form but we do not report the spectral moments of signed hand forms. Table 4 shows the spectral moments of the archimedene structure (C&). The number of self-returning walks of both signed and unsigned forms of odd lengths were found to be zero and hence not
Volume 175, number 3
7 December 1990
CHEMICAL PHYSICS LETTERS Table 4 The first 26 even spectral moments of&
Fig. 3. The vertex labellingused to generate the spectral moments of the signed foot form (Cm). Note that the graph in this figure is another topological representation of Cw found in ref. 181.
shown. This differs from the two forms of Ceo clusters. This is because Clzo graph is a bipartite graph while both forms of C6,, (fig. 1) are not bipartite. All spectral moments of the ordinary CL*,,structure are multiples of 120, the number of vertices in Clzo while this is in general not the case for the signed graph. However a signed graph with a particular labelling could be found to satisfy this condition. Note that since C,zo structure is a bipartite graph, there exists a canonical labelling such that only vertices with odd labels are connected to even labels and vice versa.
4. Conclusion In this article we obtained the walk generating functions and the spectral moments of the signed and ordinary foot and the ordinary hand form of C60as well as archimedene (CLzo). The WGFs of all these structures can be derived exactly while the spectral moments form more complex sequence of numbers. The spectral moments were found to discriminate the topological difference in the foot and hand forms of Cdo. However, up to fourth spectral moment, all spectral moments of the foot and hand forms are identical. The spectral moments and walk generating
(fig. 2) ‘)
360 2040 13560 97080 727800 5627160 44459880 356945400 2901704280 23829404760 197369138040 1646764076160 13828140774600 116774056575480 991066503921240 8448848191002360 72314836069731960 621178325061229080 5353166258304475560 46267582951755888120 400956703 116911720280 3483134534672414441880 30325309810761615820920 264559569117 174676305240 2312359319575042657009800 20246000581064962255021560 ‘) Note that all odd spectral moments are zero since ClzOgraph is a bipartite graph and hence are not shown.
functions of signed graphs were found to be dependent on labelling of vertices. Therefore, it would be interesting to explore if there is a labelling of the signed graph such that the even spectral moments of the signed graphs differ only in the signs compared to the corresponding moments of ordinary graphs. It appears that for only bipartite graphs such as Clzo this may be possible. It would also be of .interest to seek signed graphs with maximal and minima number of even self-returning walks. Such studies could be the topic of future investigations.
Acknowledgement The author thanks the referees for many useful comments.
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CHEMICAL PHYSICS LETTERS
References [l] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. Smalley, Nature 318 (1985) 162. [2] Y. Liu, S.C. O’Brien, Q. Zhang, J.R. Heath, F.K. Tittle, R.F. Curl, H.W. Kroto and R.E. Smalley, Chem. Phys. Letters 126 (1986) 215. [3] J.R. Heath, SC. O’Brien, Q. Zhang, Y. Liu, R.F. Curl, H.W. Kroto, F.K. Tittle and R.E. Smalley, J. Am. Chem. Sot. 107 (1985) 7779. [4] A.J. Stone and D.J. Wales, Chem. Phys. Letters 128 (1986) 501. [ 51D.J. Klein, T.G. Schmalz, G.F. Hite and W.A. Seitz, Nature 323 (1986) 703. [ 61 D.J. Klein, W.A. Seitz and T.G. Schmalz, Chem. Phys. Letters 130 (1986) 203. [7] A.D.H. Haymet, Chem. Phys. Letters 122 (1985) 421. [ 81 R.A. Davidson, Theoret. Chim. Acta 58 (1981) 193. [ 91 V. Elser and R.C. Haddon, Nature 325 ( 1987) 792. [lo] R.B. Mallion, Nature 325 (1987) 700. [ 111 T. Shibuya and M. Yoshitani, Chem. Phys. Letters 137 (1987) 13. [ 121 K. Balasubramanian and X.Y. Liu, J. Comput. Chem. 4 (1988) 406.
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[ 131 K. Balasubramanian and X.Y. Liu, Intern. J. Quantum Chem. Symp. 22 (1988) 319. [ 141J. Dias, J. Chem. Educ. 66 (1989) 1012. [ 151 M. RandiC, J. Chem. Inf. Comput. Sci. 18 (1978) 101. [ 16) M. Rand&J. Comput. Chem. 1 (1980) 386. [ 171 M. Rand& G.M. Brissey, R.B. Spencer and C.L. Wilkins, Computer Chem. 3 (1979) 5. [ 181 M. RandiC and CL. Wilkins, J. Chem. Inf. Comput. Sci. 20 (1980) 36. [ 191 F.T. Wall and D.J. Klein, Proc. Natl. Acad. Sci. US 76 (1979) 1529. 120‘I R.A. Marcus, J. Chem. Phys. 43 (1965) 2643. [211E.W. Montroll, in: Applied combinatorial mathematics, ed. E.F. Beckenbach (Wiley, New York, 1964) pp. 96-143. 122 :I K. Balasubramanian, Computer Chem. 9 ( 1985) 43. [23‘I D.M. CvetkoviC, M. Doob and H. Sachs, Spectra of graphs (Academic Press, New York, 1986). ~24,IJ.K. Burdett, in: Graph theory and topology in chemistry, eds. R.B. King and D.H. Rouvray (Elsevier, Amsterdam, 1987) pp. 302-324; J.K. Burdett and S. Lee, J. Am. Chem. Sot. 107 (1985) 3050. [ 251 R.B. King, Theoret. Chim. Acta 56 ( 1980) 269. [ 261 R.B. King, Theoret. Chim. Acta 44 ( 1977) 233. [ 271 K. Balasubramanian, J. Comput. Chem., in press. [ 283 F. Harary, Graph theory (Addison-Wesley, Reading, 1969). [ 291 L. Peusner, Appl. Discrete Math. 19 ( 1988) 305.
CHEMICAL PHYSICS LETTERS
7 December 1990
Spectral moments and walks for large carbon cage clusters K. Balasubramanian
’
Department of Chemistry, Arizona State University, Tempe, AZ 85287-1604, USA Received 1 August 1990; in final form 6 September 1990
Spectral moments and walks for two forms of the Cso cluster (football form and hand form) as well as a form of ClzOcluster are obtained. It is shown that spectral moments and other topological characteristics of these clusters can be readily obtained using vector&d computer codes.
1. Introduction Ever since Smalley and coworkers [ l-31 have generated the unusually stable Cho cluster buckminsterfullerene (footballene), there have been numerous experimental and theoretical studies on large carbon cage clusters [4-141. In addition to the aesthetic appeal of the football molecule, the generation of this molecule has particularly stimulated graph theoretical and other topological studies in the characterization of large cluster graphs of icosahedral and dodecahedral symmetries. The spectral moments of a graph are fundamental structural invariants (although not unique) and are not only useful in the topological characterization of the connectivity networks hidden in the structure but also in numerous other applications ranging from enumeration of self-returning walks (the famous P6lya problem ) , rapid computation of energy eigenvalues through spectra moments to molecular coding, etc. [ 15-261. Burdett and Lee [ 241 have shown that important insights into similarity of energy difference curves of molecules can be obtained through the various spectral moments of the associated molecular graphs. The set of all spectral moments of a molecular structure can be utilized to construct the electronic density of states. If two graphs have the same set of spectral moments up to mth moment, then it is proposed that only moments of order greater than m will be im’ Camille and Henry Dreyfus Teacher-Scholar.
ponant for the energy difference curves [24]. The energy difference curves obtained from the spectral moments are in turn useful to comprehend the structure of molecules to solid-state materials. The spectral moments of the carbon cluster cages have not been obtained up to now although the eigenvalues and the characteristic polynomials have been obtained before [ 12,131. The objective of this Letter is to show that matrix power methods can be efficiently harnessed to obtain spectral moments and walks on large carbon clusters.
2. Method of enumerating walks and spectral moments A walk on a graph is a sequence of edges a random walker can traverse continuously starting on any vertex and ending on any vertex including the possibility of returning to a vertex visited before during the walk. A self-returning walk is defined as a walk in which the random walker returns to the starting vertex at the end of the walk. The length of a walk is the number of edges traversed in a walk. The kth spectral moment of a graph, S, can be shown to be the number of self-returning walks of length k (for additional discussions on walks and spectral moments see refs. [ 16,17,22]). The walk generating function (GF) Wand the selfreturning walk generating function SRW can be defined as
0009-2614/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
273
Volume 175, number 3
CHEMICAL PHYSICS LETTERS
5 Nkxk,
W=
k-0
SRW= i &Xk, k=O
where Nk is the number of possible walks of length k in the given graph and S, is the number of self-returning walks (spectra moment) of length k. A vectorizcd computer code to generate the walks and spectral moments of graphs containing 200 or larger number of vertices has been developed [ 27 1. This code is based on the matrix power method. It is based on a theorem [ 28 ] that the sums of powers of the adjacency matrices enumerate walks. In symbols, Nk= C (Ak)ij, ij &= c (Ak),, , i
where Ak is the kth power of the adjacency matrix. There are other techniques available for deriving walks especially ones based on the characteristic polynomials and spectra (see refs. [ 23,241 for details). But the method used here requires no modification for directed and signed graphs. Therefore, computation of various powers of the adjacency matrices of graphs in the above method is required to generate spectral moments and walk GFs. In general the walk GFs can propagate up to infinite order but it suff%zes to stop the evaluation for k= n since.higher walks can be enumerated using the Cayley-Hamilton theorem. Also, we omit the constant term (number of vertices in the graph) in Wand SRW. We also compute spectral moments and walks on the signed form of C6,,_The adjacency matrix of a signed graph is defined as A?)=
0 1 = - 1 = 0
=
introduce directionality to walks thereby yielding effective walks including directions. For several directionally oriented properties such as magnetism, we envisage signed graphs to be useful. A disadvantage with the use of signed graphs is that the spectral moments and walks of signed graphs are dependent on labelling of vertices, in general. For bipartite graphs however, a canonical labelling is possible. For bipartite graphs vertices can be labelled such that odd vertices are connected only to even vertices and vice versa.
3. Results and discussions Fig. 1 shows two forms of Cho clusters called colloquially the foot form and the hand form (see refs. [ 11,121). The foot form is the famous buckminsterfullerene while the hand form is not expected to be as stable due to not only the presence of numerous three-membered rings but also due to the absence of six-membered rings. Fig. 2 shows a structure for the ClzO cluster called the archimedene proposed by Haymet [ 71. We do not expect this form of Clzo to
_-d,
; - _ \.
..;----:
-.I
;’
’ ,
:
,‘: -.-A
?, -r-q*
.:
i_ ‘.
,____; .’ \ I 1
@
. ‘. .
HPNO
Fig.
I. The foot and band forms of C6,,.
ifi=j,
if i # j, i and j are connected and i > j , if i # j, i and j are connected and i c j , otherwise.
The signed graphs appear in many applications including electromagnetic theory and the theory of relativity [ 29 1. Signed graphs characterize additional topological information when ordinary graphs may fail to differentiate these. Furthermore signed graphs 274
7 December 1990
Fig. 2. The archimedene cluster (C120).
Volume
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PHYSICS
be unusually stable due to four-membered rings but it is conceivable that this structure for CLzOis a viable possibility. Tables 1 and 2 show the walk GFs and spectral moments of the ordinary foot and hand forms, respectively of CeO,while table 3 shows the results of the signed form C6,, (foot). We used quadruple arithmetic precision to generate the walks and spectral moments. This is required for these graphs since round off errors tend to propagate. It can be easily shown that for all graphs which contain vertices with the same degree (valence) d, the walk GF is given by v31
540 4860 43740 393660 3542940 31886460 286978140 2582803260 23245229340 209207064060
1990
Table 2 The first 58 spectral moments of the hand form of C, (fig. 1)
Table 1 Walks and spectral moments of the foot form of Cso first 20 coefficients in the walk GF 180 1620 14580 131220 1180980 10628820 95659380 860934420 7748409780 69735688020
7 December
LETTERS
0
180
120 1200 10080 80760
900 5340 35460 253620
640200 5093400 40898880
1905660 14811060 117929220
332022240 2724768720 22584987720 188865450360
956060220 7859799300 65340031860 548164718940
1591637358600 13503869994000 115244202689040 988576437445440
4633975769460 3942832 1305860 337353046060380 2900496236873220
8518588872039960 73700566087539240 639933146362108920
25044939940062900 217080643840389180 1888005056872838580
5574450244132047840 48701071093645377600 426606137649840554160
144093969478771638780
3745976162778151137000 32965588082310403244760 290691572663296782772200 2568061851526963237049520 22725431519492831691481200 201414299786386108356241440 787651523135931276819836600
16471000071368648580 1263778538627781666180 11109603786476901689460 97868290409323698899100 863819840896828376802420 7637849892658549126069380 67642704630179032131725340 599947394048498432130601860 5328352422241637700903475380
first 58 spectral moments 0 0 120 1680 18360 184800
180 900 5460 36420 256500 1873620
1783080 16746000 154352520 1403932800 12650599080 113259310320 1009684221720 8977561689600
14073540 108202500 848993460 6783392100 55086863220 453895602420 3788421227940 31979999137860
79711374249240 707384238610800 6278208974261400 55750270236199680
272635097633940 2344134895129860 20302537026863220
495459722454787080 4407471717951260400 39248547319074569640 349878641275493877600 3122194389896999600520 27888692364449629021200 249339995789263026526200 2231091951544049450239200 19978807833978771236840760 179024924705499736515673680 605 15 1473848047475480229240
176934325344415380 1550075679421949700 13639955804155214340 120471316578120932340 1067339198155547291940 9480855692875722930420 84398045708842207276980 752663457823581639132900 6722369998676165376954180 60115767186017862345323220 538156067456971472666165700 4821771501779283290257274100
w=
+&-,
where n is the number of vertices. The expansion of the above function yields W=n( L+du+d2x2+d3x3t...), although in our discussion we omit the constant term. Note that therefore for both the foot and hand forms of CSOthe Ws become identical and the coefficients of xk in both forms are 60 x 3k. The results obtained (only first 20 coefftcients are shown) from our computer codes in table 1 conform to this exact analytical result thereby assuring that both the inputs for these graphs and codes are free of bugs. It is interesting to note that although Ws are identical for the foot and hand form the spectral moments differ except for the first, second and fourth moments of the foot and hand forms. This is comforting since spectral moments have been proposed as molecular descriptors [ 161. We note that the spectral moments of the hand form are in general 215
Volume 175, number 3
CHEMICAL PHYSICS LETTERS
7 December 1990
Table 3 Walks and spectral moments of the signed foot form of C60 (fig. 3 ) first 60 coefficients in the walk GF
first 60 spectral moments
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-76 268 - 1284 6836 -38180 220156 - 1304460 7927372 - 49346244 314190228 -2042844772 13539102332 -91288612844 62498 1158796 -4336268097540 30436050757364 -215760769291300 1542521010952828 -11107139739600332 80463268 154550348 -585865402150167492 4283960873722605524 -31436806791348290916 231378204076767666428 - 1707200681925154806828 12622543907028171672972 -93489115188297469603076 693431455908954915510580 -5149577297325719259529508 38280682790533054885149436
equal or larger than the foot form. Thus there are more possibilities of returning to the starting vertex in the hand form. This is consistent with the connectivity of the hand form as compared to the foot form. That is, the presence of three-membered rings facilitates greater probability for the walker to return to the starting vertex in the hand form compared to the foot form. The walks and spectral moments of the signed form of Ceo with labelling in fig. 3, is shown in table 3. As noted before the walks and spectral moments of signed graphs are label dependent. However, a few terms are independent of the labels. As seen from table 3 all odd walks and spectral moments are zero. This is because the net effect of walking in the same and opposite directions cancel out leading to zero coefficients for all odd number of walks. We tested 276
-180 900 - 5076 30276 - 186900
1182708 - 7632756 50080772 - 333373428 2247905700 -15334130420 105705275700 -735630655316 5163566570628 -36525234178836 260159107586308 -1864491454057556 13435486395672564 -97283418495594164 707395586546432516 - 5162906959438829364 37803236000712441444 -277576997127292038964 2043130837567786737716 -15070410807770717802900 111364305414300031902084 -824234413674118485221460 6IO8651122305790206782404 -45326101689217744335479956 336658469605220442756149748
this out on signed CbOwith different labelling. The coefficients of odd walks were found to be independent of labelling as they vanish. The number of even walks however was found to depend on labelling. This is because for every walk of length k with k, positive steps and k_ negative steps there exists an opposite walk with k_ positive steps and k, negative steps. The effect of the former and latter walks cancel out for walks of odd length. The fifth and sixth spectral moments are the first and second for which the signed and ordinary forms of CGOdiffer. A similar behavior was noticed for the hand form but we do not report the spectral moments of signed hand forms. Table 4 shows the spectral moments of the archimedene structure (C&). The number of self-returning walks of both signed and unsigned forms of odd lengths were found to be zero and hence not
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CHEMICAL PHYSICS LETTERS Table 4 The first 26 even spectral moments of&
Fig. 3. The vertex labellingused to generate the spectral moments of the signed foot form (Cm). Note that the graph in this figure is another topological representation of Cw found in ref. 181.
shown. This differs from the two forms of Ceo clusters. This is because Clzo graph is a bipartite graph while both forms of C6,, (fig. 1) are not bipartite. All spectral moments of the ordinary CL*,,structure are multiples of 120, the number of vertices in Clzo while this is in general not the case for the signed graph. However a signed graph with a particular labelling could be found to satisfy this condition. Note that since C,zo structure is a bipartite graph, there exists a canonical labelling such that only vertices with odd labels are connected to even labels and vice versa.
4. Conclusion In this article we obtained the walk generating functions and the spectral moments of the signed and ordinary foot and the ordinary hand form of C60as well as archimedene (CLzo). The WGFs of all these structures can be derived exactly while the spectral moments form more complex sequence of numbers. The spectral moments were found to discriminate the topological difference in the foot and hand forms of Cdo. However, up to fourth spectral moment, all spectral moments of the foot and hand forms are identical. The spectral moments and walk generating
(fig. 2) ‘)
360 2040 13560 97080 727800 5627160 44459880 356945400 2901704280 23829404760 197369138040 1646764076160 13828140774600 116774056575480 991066503921240 8448848191002360 72314836069731960 621178325061229080 5353166258304475560 46267582951755888120 400956703 116911720280 3483134534672414441880 30325309810761615820920 264559569117 174676305240 2312359319575042657009800 20246000581064962255021560 ‘) Note that all odd spectral moments are zero since ClzOgraph is a bipartite graph and hence are not shown.
functions of signed graphs were found to be dependent on labelling of vertices. Therefore, it would be interesting to explore if there is a labelling of the signed graph such that the even spectral moments of the signed graphs differ only in the signs compared to the corresponding moments of ordinary graphs. It appears that for only bipartite graphs such as Clzo this may be possible. It would also be of .interest to seek signed graphs with maximal and minima number of even self-returning walks. Such studies could be the topic of future investigations.
Acknowledgement The author thanks the referees for many useful comments.
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