The 10-Cages and Derived Configurations

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Discrete Mathematics 275 (2004) 265–276 www.elsevier.com/locate/disc

The 10-cages and derived conÿgurations

T. Pisanskia;1 , M. Bobena , D. MaruÄsiÄca;2 , A. OrbaniÃca , A. Graovacb;3 aUniverza v Ljubljani, IMFM, Jadranska 19, 1000 Ljubljana, Slovenia bInstitut R. BoskoviÄ c,Ã P.O. Box 180, 10002 Zagreb, Croatia

Received 28 March 2002; received inrevised form 2 December 2002; accepted 2 January2003

Dedicated to Professor AlexandruT. Balabanonthe occasionof his 70th birthday

Abstract

Symmetry properties of the three 10-cages on 70 vertices are investigated. Being bipartite, these graphs are Levi graphs of triangle- and quadrangle-free (353) conÿgurations. For each of these graphs a Hamiltoncycle is givenvia the associated LCF notation.Furthermore, the automorphism groups of respective orders 80, 120, and 24 are computed. A special emphasis is givento the Balaban10-cage, the ÿrst knownexampleof a 10-cage (Rev. RoumaineMath. Pure Appl. 18 (1973) 1033–1043), and the corresponding Balaban conÿguration. It is shown that the latter is linear, that is, it can be realized as a geometric conÿguration of points and lines in the Euclidean plane. Finally, based on the Balaban conÿguration, an inÿnite series of linear

Keywords: Graphs; Cages; Conÿgurations

1. Introduction

The motivation for the paper is a connection between cages (graphs) and conÿgura- tions (incidence structures). In particular, the 2n-cycles ingraphs correspondto n-gons

1 Supported in part by Ministrstvo za znanost in tehnologijo Republike Slovenije, Grant J1-6161, J2-6193. 2 Supported in part by Ministrstvo za znanost in tehnologijo Republike Slovenije, Grant J1-0496. 3 Supported in part by the Ministry of Science and Technology of Croatia, Grant No. 00980606. E-mail addresses: [email protected] (T. Pisanski), [email protected] (M. Boben), [email protected] (D. MaruÄsiÄc), [email protected] (A. OrbaniÃc), [email protected] (A. Graovac).

0012-365X/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0012-365X(03)00110-9 266 T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276 in conÿgurations. We extend the present knowledge about this subject to 10-cages. Also, some additional properties of graphs and conÿgurations are studied.

1.1. Conÿgurations

A(symmetric) conÿguration (vr) is an incidence structure of v points and v lines such that there are r lines through each point, r points on each line, and two lines meet in at most one point. We can relate conÿgurations to graphs in the following sense. Let L = L(C) be the bipartite graph with v “black” vertices representing the points of the conÿguration C, with v “white” vertices representing the lines of C, and with an edge joining two vertices if and only if the corresponding point and line are incident in C. We call L the Levi graph of the conÿguration C. The following proposition characterizes (vr) conÿgurations in terms of their Levi graphs. Recall that the length of the shortest cycle ina graph G is called the girth of G.

Proposition 1. An incidence structure is a (vr) conÿguration if and only if its Levi graph is r-regular and has girth at least 6.

Proof. See [11].

With each conÿguration C the dual conÿguration Cd may be associated by reversing the roles of points and lines in C. Both C and Cd share the same Levi graph, only the black-white coloring of its vertices is reversed. If C is isomorphic to its dual we say that C is self-dual and a corresponding isomor- phism is called a duality. A duality of order 2 is called a polarity. Anisomorphism of C to itself is called an automorphism. Automorphisms of C form a group. A con- ÿguration C is called linear if it canbe realized inthe Euclideanplanewith points and (straight) lines, see [16]. We say that a conÿguration C is triangle-free if L(C) has girth at least 8, similarly we say that C is triangle- and quadrangle-free if L(C) has girth at least 10, etc. Also, a conÿguration C is called d-gonal if the girth of L(C) equals 2d. Inthis contextwe say that d is a normal number if for each integer n there exists v ¿ n such that there is a d-gonal linear (v3) conÿguration. For example, 3 is a normal number, see [16]. The search for the smallest n-gonal conÿguration is therefore equivalent to the search for the smallest bipartite graph with girth 2n. This is where we meet a well-known problem of cages from graph theory.

1.2. Cages

For aneasy introductiontothe subject the reader is referred to [ 25]. For a simple graph the girth is at least 3. The smallest trivalent graph of girth g is called a g-cage. Obviously K4 is the unique 3-cage and K3;3 is the only 4-cage. The Petersen graph P(5; 2) is the only 5-cage. The 6-cage is known as the Heawood graph; see Fig. 3(b). The unique 7-cage has 24 vertices. The 8-cage is known as the Cremona–Richmond graph or the Tutte 8-cage. There are 18 non-isomorphic 9-cages which are computed in[ 8]. T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276 267

(a) (b)

Fig. 1. The Balaban10-cage or B(5) (a). The two outer 10-gons can be interchanged with the two inner 10-gons. Its generalization B(9) is shownonthe right.

(a) (b)

Fig. 2. The other two 10-cages are also bipartite and give rise to the conÿgurations that are triangle- and quadrangle-free.

Balabanfoundoneof the three 10-cages which is showninFig. 1(a), see [1]. The other two 10-cages are showninFig. 2. Up to g 6 10 the g-cages are classiÿed completely. For g = 12 there is a unique 12-cage. All other cases (g = 11, g¿12) are still open. For a survey on cages see also [24,26,29,30]. The latest results oncages are givenin[ 23]. Cages and related graphs have found their applications in chemistry [7], e.g. in modelling chemical reactions [2] and degenerate rearrangements [18].

2. Cages and conÿgurations

2.1. Overview

We have mentioned already that the size of the smallest n-gonal symmetric (v3) conÿgurations is related to the existence of cages. By Proposition 1, each bipartite 268 T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276

5 3

1 2 4

(a) (b)

Fig. 3. Fano conÿguration (a). Its Levi graph is the Heawood graph or the 6-cage (b).

9 4 1

2 8 15 3 13 7

(a) (b)

Fig. 4. Cremona–Richmond (153) conÿguration, the smallest (v3) conÿguration without triangles (a) is linear. Its Levi graph is the Tutte 8-cage which is a Z5 covering graph over the voltage graph (b).

2n-cage represents a Levi graph of the smallest m-gon-free (v3) conÿguration for m¡n, i.e. n-gonal conÿguration (possibly a pair of dual conÿgurations). The cages with an odd girth are not bipartite graphs and will not attract our attention. But, interestingly, all known n-cages where n is anevenintegerare bipartite graphs and hence generate n=2-gonal conÿgurations for n ¿ 6. The only 6-cage, the Heawood graph, has 14 vertices. It gives the smallest con- ÿguration, the Fano (73) conÿguration or the projective plane of order 2, see Fig. 3. The Tutte 8-cage is a bipartite graph on30 vertices. It representsa Levi graph of the smallest (v3) conÿguration without triangles, the Cremona–Richmond (153) conÿgura- tion[ 11], see Fig. 4. The Heawood graph and the Tutte 8-cage contain a Hamilton cycle and are highly symmetric. It is well known that the Fano conÿguration is not linear while the Cremona–Richmond conÿguration is. Both are self-dual conÿgurations. T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276 269

The remaining part of the article is devoted to the investigation of these properties in the case of 10-cages and the corresponding conÿgurations.

2.2. Ten-cages

As mentioned in the introduction, there are three non-isomorphic 10-cages. They are all bipartite graphs on70 vertices. Let us ÿrst investigate the existence of a Hamilton cycle. For cubic graphs with a givenHamiltoncycle there is a useful notationdueto Frucht et al. [ 12]. All we have to do is to list the lengthsof chords measured injumps whenwe traverse the vertices along the Hamilton cycle. Such a list is called the LCF notation. For instance, K4 canbe described by the sequence[2 ; 2; 2; 2] which canbe shortenedto [2 4]. The LCF notation for the Tutte 8-cage is [(−13; −9; 7; −7; 9; 13)5]. The LCF codes for the 10-cages are, respectively, [ − 9; −25; −19; 29; 13; 35; −13; −29; 19; 25; 9; −29; 29; 17; 33; 21; 9; −13; −31; −9; 25; 17; 9; −31; 27; −9; 17; −19; −29; 27; −17; −9; −29; 33; −25; 25; −21; 17; −17; 29; 35; −29; 17; −17; 21; −25; 25; −33; 29; 9; 17; −27; 29; 19; −17; 9; −27; 31; −9; −17; −25; 9; 31; 13; −9; −21; −33; −17; −29; 29];

[(−29; −19; −13; 13; 21; −27; 27; 33; −13; 13; 19; −21; −33; 29)5]; [9; 25; 31; −17; 17; 33; 9; −29; −15; −9; 9; 25; −25; 29; 17; −9; 9; −27; 35; −9; 9; −17; 21; 27; −29; −9; −25; 13; 19; −9; −33; −17; 19; −31; 27; 11; −25; 29; −33; 13; −13; 21; −29; −21; 25; 9; −11; −19; 29; 9; −27; −19; −13; −35; −9; 9; 17; 25; −9; 9; 27; −27; −21; 15; −9; 29; −29; 33; −9; −25]: This proves the following proposition.

Proposition 2. Each 10-cage contains a Hamilton cycle.

An important part of interest when speaking about graph properties is an existence of symmetry. If the automorphism group of a graph G contains a semiregular element (ÿxing the two sets of a bipartition), then G is called a polycirculant. Part of the motivationfor studyingthis class of graphs is a problem proposed in1981 by the third author who asked if there exists a vertex-transitive graph without a nonidentity semiregular automorphism [19, Problem 2.4], that is, a vertex-transitive graph which is not a polycirculant. This problem attracted a wider interest in the mathematical com- munity when it was reproposed in a slightly more general group-theoretic setting. The more general form, due to Klin [9, Problem BCC15.12], asks whether every transitive 2-closed permutation group contains a ÿxed-point-free element of prime order. The term elusive has recently been adopted for a transitive ÿnite permutation group without a non-trivial semiregular subgroup. As was shown in [10], there are inÿnitely 270 T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276

(a) (b)

Fig. 5. The smallest 5-gonal (v3) conÿguration (a) resulting from the Balaban 10-cage. It is a covering graph over (b) with group Z5. many elusive transitive permutation groups, but none of them is 2-closed. It is believed that no 2-closed transitive group is elusive. This conjecture is usually referred to as the polycirculant conjecture. For further results and recent advances with regards to this conjecture see [10,13,19,20,22]. The concept of polycirculants extends naturally to conÿgurations (see the next subsection). By computing the automorphism groups of all 10-cages we get the following result.

Proposition 3. The automorphism groups of the 10-cages have orders 80; 120; 24, respectively. The Balaban cage and the second 10-cage are polycirculants.

The concept of voltage graphs is generally used to simplify the description of large graphs. We refer the reader who is not familiar with this topic to the book by Gross and Tucker [14] or to the book by White [28]. For example, the Tutte 8-cage is a Z5 covering graph over the voltage graph in Fig. 4(b). The ÿrst and the second 10-cage are Z5 covering graphs over the voltage graphs in Figs. 5(b) and 6(c). Note that in the pictures we present, voltages and arrows are omitted from the edges carrying voltage 0.

2.3. The smallest 5-gonal (v3) conÿgurations

The smallest 5-gonal conÿgurations, triangle- and quadrangle-free conÿgurations, result from 10-cages. The 5-gonal (353) conÿgurationgivenbythe Balabancage is self-polar andis presented in Fig. 5(a). We call it the Balaban conÿguration. The second cage gives rise to a pair of dual 5-gonal (353) conÿgurations, see Figs. 6(a) and (b). The third cage also gives rise to a pair of dual conÿgurations. Hence we have the following result. T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276 271

(a) (b)

(c)

Fig. 6. The second 10-cage is a Z5 covering graph over voltage graph (c) and gives rise to a pair of dual linear (353) conÿgurations (a) and (b).

Proposition 4. There are 5 non-isomorphic 5-gonal (353) conÿgurations. Also, these are the smallest 5-gonal (v3) conÿgurations.

Another important concept which has been extensively studied for over a century is the existence of a realization of conÿgurations in the Euclidean plane, that is, the question whether a conÿguration is linear or not.

Proposition 5. All ÿve 5-gonal (353) conÿgurations are linear.

Proof. The coordinates can be found in [4].

The problem of ÿnding a realization can be further extended to the search for nice realizations. There is no general theory, but the use of symmetry of their Levi graphs proves to be helpful. A conÿguration C is polycyclic if there exists anautomorphism which is semiregular, meaning that all orbits on points and lines are of the same size. If has order k then C is called a k-cyclic conÿguration. The corresponding Levi graph of a polycyclic conÿguration is a polycirculant. There is a theory developed in[ 5] that enables one to construct the so-called rotational drawings of many conÿgurations admitting a polycyclic structure. In this way the realizations in Figs. 4–6 were obtained. See also [15,17,21] for related concepts regarding conÿgurations and graphs which possess certain symmetries. 272 T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276

Table 1 The construction sequence for the coordinates of a conÿguration arising from the third 10-cage; in order to obtaina realization,the parameters i,06 i 6 27, must satisfy the equation[1 ; 2; 34]=0

T T x32 =(1; 0; 0) , x33 =(0; 1; 0) , T T x34 =(0; 0; 1) , x35 =(1; −1; 1) , x31 = x32 + x3326 + x3427, x30 = x32 + x3324 + x3425, x29 = x32 + x3322 + x3423, x27 = x32 + x3320 + x3421, x25 = x32 + x3318 + x3419, x23 = x32 + x3316 + x3417, x22 = x32 + x3314 + x3415, x20 = x32 + x3312 + x3413, x19 = x32 + x3310 + x3411, x16 = x32 + x338 + x349, x15 = x32 + x336 + x347, x12 = x32 + x334 + x345,

x8 = x32 + x332 + x343,

x26 = x27 + x331, x5 = x25 + x150, x28 = [29; 35; 32]x22 + [35; 29; 22]x32, x24 = [31; 23; 29]x25 + [23; 31; 25]x29, x21 = [34; 22; 26]x25 + [22; 34; 25]x26, x18 = [31; 12; 35]x19 + [12; 31; 19]x35, x17 = [32; 16; 26]x18 + [16; 32; 18]x26, x14 = [19; 15; 23]x22 + [15; 19; 22]x23, x13 = [33; 14; 30]x29 + [14; 33; 29]x30, x11 = [34; 12; 16]x15 + [12; 34; 15]x16, x10 = [27; 11; 30]x20 + [11; 27; 20]x30, x9 = [35; 8; 23]x10 +[8; 35; 10]x23, x7 = [16; 8; 21]x20 +[8; 16; 20]x21, x6 = [31; 7; 28]x27 +[7; 31; 27]x28, x4 =[9; 5; 13]x12 +[5; 9; 12]x13, x3 = [32; 4; 20]x19 +[4; 32; 19]x20, x2 =[3; 24; 33]x8 + [24; 3; 8]x33, x1 =[5; 6; 30]x17 +[6; 5; 17]x30

The Balabanconÿgurationispolycyclic. Its rotationalrealizationis shown inFig. 5(a). It results from the symmetry givenby the descriptionof its Levi graph using the voltage graph in Fig. 5(b). The same is true for the two dual conÿgurations given by the second 10-cage. Their rotational realizations are depicted in Figs. 6(a) and (b). The two 5-gonal (353) conÿgurations given by the third cage are also linear, but they are not polycyclic and we cannot give rotational realizations as we did in the previous cases. Nevertheless, their realizability canbe easily proved usingthe algebraic methods described in[ 6]. The coordinates xi,16 i 6 35, for one of them can be obtained from the constructionsequencegiveninTable 1. Here we use the bracket notation [i; j; k] for det(xi; xj; xk ). See [6] for more details. The values for parameters i,06 i 6 27, canbe determinedinsuch way that the vectors xi represent a realization over rational numbers, see Table 2.

2.4. Generalizations

Another advantage of using voltage graphs for a description of graphs is a possibility of constructing a sequence of graphs and conÿgurations having the same properties.

Theorem 6. There exists an inÿnite sequence of 5-gonal linear (v3) conÿgurations. Hence,5is a normal number.

Proof. Let B(n) be a covering graph of the base graph in Fig. 5(b) with group Zn. All graphs B(n) are bipartite. If n ¿ 5 is odd thenit is easy to see that the girth of T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276 273

Table 2 The set of rational parameters 0;1;:::;27 for the construction of coordinates presented in Table 1 which gives a rational realization of the conÿguration determined by the third 10-cage, see [4]

0= −2574943097082595345544340065044256416763150642661777434707789 59667876208778207177019482496817757205402642189156706725346890 22622140368112040532401975002792847586056247186185267095609210 72224023908305523173073258831927707060390209692692544175062521 00867567334856330270701885399779443847630494944045197210113833 45686581979031244339659271649948087846117980910301123388413720 57051815992941513500155776031972956537535643382299035992535042 673717153882602556709161655598342768196720660= 85664207833558481307133002407180419053546312271907618909401096 80544957921299249445923041130506627139497716141753909248899902 78029899996598841756184775905246722644857221403556605555175859 46369322765272707716115142694797498080293757811603542109067791 07289460236357135937824869454820173030487000130136964879208257 80308132775211077977349189135263667885925246786092378159999780 63414018873915609490401927093884298498944747578152115900107889 059442671841249813781870735761562299499741503;

1= −7123834754731184528402693911971923451330058149875518104914623 48056953430094500547074853487247119654330646982662266593158414 49027401729823664422731097278262026340204310263333500332789641 803738213536180557379054243= 12788224222663276361356042073165585429965807349185057796106157 07279731211934682828012940882657221804428098394679075529224787 98445062608461161721810937141364826359769343341366263271236295 4440098660148871267133974;

2= −3050903110258722487817260103307920379457268495239756349868330 06626736546822678992457493981425096763980125936846850431582649 0179113374985805683= 80499635231414144477909109832276161549112142353787505163054529 96573531181800497472715753121277887854626212813887682347433470 5127358277359656;

3= −9971768468591115756610251558360481907= 151463727449019180310424328127048392;

4 =81;5 = −73;6 =56;7 =98;8 =33;9 =71;10 = −62,

11 = −87;12 = −35;13 = −55;14 =68;15 =69;16 =73;17 = −29,

18 = −73;19 =87;20 =91;21 =73;22 =49;23 =25;24 = 28,

25 = 100;26 =24;27 = −13

the covering graph is at least 10. Thus, they determine a sequence of 5-gonal ((7n)3) conÿgurations Cn for n =5; 7; 9;::: . The ÿrst conÿgurationintheseries is the Balaban conÿguration, the third one is shown in Fig. 7. To see that all conÿgurations Cn inthe series are linearwe must ÿrst check whether the necessary condition (for rotational realization) from the algorithm given in [5] can be satisÿed by some set of real parameters for any Cn. After, for example, choosing the 274 T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276

Fig. 7. The third conÿguration in the inÿnite series of 5-gonal self-polar (v3) conÿgurations having B(9) as its Levi graph. values 1; 2;:::;8 for the ÿrst eight of nine parameters, we get in our case the equation 2 4 6 8 x2 1476 + 8120 cos + 2936 cos + 1460 cos + 1952 cos n n n n 12 2 4 + 312 cos + 1435 sin + 5515 sin n n n 6 8 12 + 5515 sin − 1004 sin − 2439 sin n n n 2 4 6 + x −20616 − 229 098 cos − 6462 cos − 19 746 cos n n n 8 12 2 4 − 37 542 cos − 46 200 cos − 15 561 sin − 125 673 sin n n n n 6 8 12 − 125 673 sin + 35 544 sin + 51 105 sin n n n 2 4 6 − 305 100 − 360 018 cos + 1 344 474 cos + 1 344 474 cos n n n 8 2 4 6 − 54 918 cos − 256 284 sin + 443 412 sin + 443 412 sin n n n n 8 −256 284 sin =0: n T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276 275

The discriminant of this quadratic equation is greater than zero for all n ¿ 5. This ensures the existence of real solutions for n ¿ 5 and the fulÿllment of the necessary condition for rotational realization. The fact that this condition is suPcient (that is, that no two points coincide) can be proved by computer calculations.

3. Conclusion

In the text we already mentioned an interesting fact that all known even cages are bipartite graphs. Therefore it makes sense to set the following conjecture.

Conjecture 7. All g-cages where g is anevenintegerare bipartite graphs.

This conjecture is known to be true for even g up to 12. This is the largest number for which the cage problem is solved. The unique 12-cage has 126 points. It is a Levi graph of two dual (633) conÿgurations which are also known as the generalized hexagons, see [27]. At this point we should also mention the enumeration results for (v3) conÿgura- tions, v 6 18, in[ 3] obtained by computer calculations. The authors report that the results were extended to v = 19 after long parallel computer runs. The number of all non-isomorphic (193) conÿgurations is 7 640 941 062. Among them there are 1 992 044 self-dual conÿgurations, 1 991 320 self-polar conÿgurations, 3 cyclic conÿgurations, 14 triangle-free conÿgurations. The number of triangle-free conÿgurations is known up to v = 21. The gap between the computer approach and theoretical considerations seems to be unsurmountable which makes the conjecture attractive.

References

[1] A.T. Balaban, Trivalent graphs of girth nine and eleven and relations among cages, Rev. Roumaine Math. Pure Appl. 18 (1973) 1033–1043. [2] A.T. Balaban, Reaction Graphs, in: D. Bonchev, O. Mekenyan (Eds.), Graph Theoretical Approaches to Chemical Reactivity (Understanding Chemical Reactivity), Vol. 9, Kluwer Academic Publishers, Dordrecht, 1994. [3] A. Betten, G. Brinkmann, T. Pisanski, Counting symmetric conÿgurations v3, Discrete Appl. Math. 99 (2000) 331–338. [4] M. Boben, T. Pisanski, Coordinatizations of the conÿgurations arising from the 10-Cages, http://www.ijp.si/Balaban/balweb.nb. [5] M. Boben, T. Pisanski, Polycyclic conÿgurations, European J. Combin. 24 (2003) 431–457. [6] J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, in: Lecture Notes in Mathematics, Vol. 1355, Springer, Heidelberg, 1989. [7] D. Bonchev, O. Mekenyan, Graph Theoretical Approaches to Chemical Reactivity (Understanding Chemical Reactivity), Vol. 9, Kluwer Academic Publishers, Dordrecht, 1994. [8] G. Brinkmann, B.D. McKay, C. Saager, The smallest cubic graph of girth 9, Combin. Probab. Comput. 5 (1995) 1–13. [9] P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference; Discrete Math. 167/168 (1997) 605–615. [10] P.J. Cameron, M. Giudici, G.A. Jones, W.M. Kantor, M.H. Klin, D. MaruÄsiÄc, L.A. Nowitz, Transitive permutationgroups without regular subgroups, J. LondonMath.Soc. 66 (2) (2002) 325–333. 276 T. Pisanski et al. / Discrete Mathematics 275 (2004) 265–276

[11] H.S.M. Coxeter, Self-dual conÿgurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950) 413– 455. [12] R. Frucht, A canonical representation of trivalent Hamiltonian graphs, J. Graph Theory 1 (1977) 45– 60. [13] M. Giudici, Quasiprimitive groups with no ÿxed, point free elements of prime order, J. London Math. Soc. 67 (2) (2003) 73–84. [14] J.L. Gross, T.W. Tucker, Topological Graph Theory, Wiley Interscience, New York, 1987. [15] B. Grunbaum,S Astral (nk ) conÿgurations, Geombinatorics 3 (1993) 32–37. [16] B. Grunbaum,S Special Topics in Geometry—Conÿgurations, Math. 553B, University of Washington, Seattle, Spring, 1999. [17] B. Grunbaum,S Astral (n4) conÿgurations, Geombinatorics 9 (2000) 127–134. [18] D.J. Klein, A. Graovac, N. TrinajstiÃc, Z. MihaliÃc, Excitation spectra for degenerate rearrangements, J. Mol. Struct. 341 (1995) 157–164. [19] D. MaruÄsiÄc, Onvertex symmetric digraphs, Discrete Math. 36 (1981) 69–81. [20] D. MaruÄsiÄc, Vertex symmetric graphs and digraphs of order pk , Ann. Discrete Math. 27 (1985) 115– 128. [21] D. MaruÄsiÄc, T. Pisanski, Weakly ag-transitive conÿgurations and half-arc-transitive graphs, European J. Combin. 20 (1999) 559–570. [22] D. MaruÄsiÄc, R. Scapellato, Permutation groups, vertex-transitive graphs and semiregular automorphisms, EuropeanJ. Combin.19 (1998) 707–712. [23] B.D. McKay, W. Myrvold, J. Nadon, Fast backtracking principles applied to ÿnd new cages, Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, January 1998, pp. 188–191. [24] M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (1999) 137–146. [25] T. Pisanski, M. RandiÃc, Bridges between Geometry and Graph Theory, in: C.A. Gorini (Ed.), Geometry at Work, MAA Notes, Vol. 53, Mathematical Association of America, Washington, DC, 2000, pp. 174–194. [26] G. Royle, http://www.cs.uwa.edu.au:80/∼gordon/data.html. [27] A. Schroth, How to draw a hexagon, Discrete Math. 199 (1999) 167–171. [28] A.T. White, Graphs of Groups on Surfaces, in: North-Holland Mathematics Studies, Vol. 188, North-Holland, Amsterdam, 2001. [29] P.K. Wong, Cages—a survey, J. Graph Theory 6 (1982) 1–22. [30] P.K. Wong, On the smallest graphs of girth 10 and valency 3, Discrete Math. 43 (1983) 119–124.