Filling in pattern designs for incomplete pairwise comparison matrices: (quasi-)regular graphs with minimal diameter
FFilling in pattern designs for incomplete pairwise comparison matrices: (quasi-)regular graphs withminimal diameter
Sándor Bozóki , , Zsombor Szádoczki , , Hailemariam Abebe Tekile June 3, 2020 Research Laboratory on Engineering & Management IntelligenceInstitute for Computer Science and Control (SZTAKI)Budapest, Hungary Department of Operations Research and Actuarial SciencesCorvinus University of Budapest Department of Industrial EngineeringUniversity of Trento, Italy
Abstract
Multicriteria Decision Making problems are important both for individuals and groups.Pairwise comparisons have become popular in the theory and practice of preferencemodelling and quantification. We focus on decision problems where the set of pairwisecomparisons can be chosen, i.e., it is not given a priori. The objective of this paperis to provide recommendations for filling patterns of incomplete pairwise comparisonmatrices (PCMs) based on their graph representation. Regularity means that eachitem is compared to others for the same number of times, resulting in a kind of sym-metry. A graph on an odd number of vertices is called quasi-regular, if the degree ofevery vertex is the same odd number, except for one vertex whose degree is larger byone. If there is a pair of items such that their shortest connecting path is very long,the comparison between these two items relies on many intermediate comparisons,and is possibly biased by all of their errors. Such an example was found in Tekile(2017) where the graph generated from the table tennis players’ matches included a a r X i v : . [ c s . D M ] M a y ong shortest path between two vertices (players), and the calculated result appearedto be misleading.If the diameter of the graph of comparisons is low as possible (among the graphsof the same number of edges), we can avoid, or, at least decrease, such cumulatederrors.The aim of our research is to find graphs, among regular and quasi-regular ones,with minimal diameter. Both theorists and practitioners can use the results, given inseveral formats in the appendix: graph, adjacency matrix, list of edges. Multicriteria Decision Making is a really important tool both at an individual and at anorganizational level. We can almost think about any kind of ranking of alternatives orweighting of criteria, like tenders, selection among schools or job offers, selection amongthe evaluation of different projects in an enterprise, etc.One of the most commonly used technique in connection with the Multicriteria DecisionMaking is the method of the pairwise comparison matrices (Saaty, 1980). One can applythis technique both for determining the weights of the different criteria and for the ratingof the alternatives according to a criterion. Usually we denote the number of criteria oralternatives by n , which means the pairwise comparison matrix is an n × n matrix oftendenoted by A . In this case the ij -th element of the A matrix, a ij shows how many timesthe i -th item is larger/better than the j -th element.Formally, matrix A is called a pairwise comparison matrix (PCM) if it is positive( a ij > for ∀ i and j ) and reciprocal ( /a ij = a ji for ∀ i and j ) (Saaty, 1980), which alsoindicates that a ii = 1 for ∀ i .When some elements of a PCM are missing we call it an incomplete PCM. There couldbe many different reasons why these elements are absent, some data could have been lostor the comparisons are simply not possible (for instance in sports (Bozóki et al., 2016)).The most interesting case for us is when the decision makers do not have time, willing-ness or the possibility to make all the n ( n − / comparisons.In this article we would like to show which comparisons are the most important ones tobe made, or more precisely what pattern of comparisons are recommended to be made inorder to get the best approximation of the decision makers’ original preferences in differentcases, when we have some assumptions on our Multicriteria Decision Making (MCDM)2roblem. The graph representation of the pairwise comparisons is a natural and convenienttool to examine our question, thus we will use this throughout the paper.Special structures from incomplete pairwise comparison matrices include (i) spanningtree, in particular if one row/column is filled in completely (its associated graph is the stargraph) (ii) two rows/columns are filled in completely (its associated graph is the union oftwo star graphs) Rezaei (2015) (iii) more or less regular graphs, for example in case of sportcompetitions Csató (2013), where the number of matches played equals for every player orteam, at least in the first phase (before the knockout phase).These examples do not take the diameter into consideration, and the first two exampleslack regularity, too. Regularity means that each item is compared to others for the samenumber of times (if the cardinality of the items to compare is odd, one of the degrees canbe smaller or greater - in our analysis, greater - by one), resulting in a kind of symmetry.In the set of connected graphs, diameter can be considered as a measure of closeness, or astronger type of connectedness.To understand the used methodology we have to define some basic mathematical con-cepts, which takes place in the second section of the article. Later on we assume thatwe know the number n of alternatives or criteria, it is also a key assumption through ourpaper that the graph that representing the MCDM problem is k -regular and we also know(or with the help of the other inputs we can determine) the diameter d of the graph. Inthe third section we provide a systematic collection of suggested incomplete pairwise com-parisons’ patterns with the help of the above-mentioned inputs and the/some graphs forthe examined cases. In Section 4 we make our conclusion and provide further researchquestions closely connected to the discussed topic.Note that regular graphs can have large diameter, e.g., a cycle on n vertices is 2-regular and has diameter d = (cid:98) n/ (cid:99) . The star graph, mentioned among the examples, hasminimal diameter 2, but it is far from being regular. Our aim is to find the graphs, among(quasi-)regular ones, with minimal diameter. We are especially interested in the smallestnontrivial values of the diameter, namely d = 2 and d = 3 . The graph representation of paired comparisons has already been used in the 1940s (Kendalland Smith, 1940). Of course after the widespread application of PCMs and incomplete3CMs it has become a really common method in the literature, see for instance Blanqueroet al. (2006), Csató (2015) or Gass (1998).Usually in these articles the authors use directed graphs for the representation, becausethey distinguish the preferred item from the less preferred one in every pair. In our ap-proach the only important thing is the following: is there a comparison between the twoelements or not. This means that we use undirected graphs, where the vertices denote thecriteria or the alternatives. There is an edge between two vertices only if the decision mak-ers made their comparison for the two respective items, while there is not an edge betweentwo vertices only if the decision makers have not made their comparison for the two units(the respective element of the PCM is missing). In order to understand the concepts sofar, there is a small example below:
Example 1
Let us assume that there are 4 criteria ( n = 4 ) and our decision maker alreadyanswered some questions, denoted their locations in the matrix by • and their reciprocalvalues by ◦ , which lead to the following incomplete PCM: A = • •◦ •◦ •◦ ◦ This incomplete PCM is represented by the graph in Figure 1.
Figure 1: Graph representation example4 s we can see there is no edge between the first and the fourth vertices, where the PCMhas missing values and there is no edge between the second and third vertices, where thesituation is the same. There is an edge between every other pair, where we have no missingvalues in the PCM.
We assume that the representing graphs are connected and k -regular through our paper,thus we need some definitions to make these concepts clear. Definition 1 (Connected graph)
In an undirected graph, two vertices u and v are calledconnected if the graph contains a path from u to v . A graph is said to be connected if everypair of vertices in the graph is connected. Definition 2 ( k -regular graph) A graph is called k -regular if every vertex has k neigh-bours, which means that the degree of every vertex is k . Definition 3 ( k -quasi-regular graph) A graph is called k -quasi-regular if exactly onevertex has degree k + 1 , and all the other vertices have degree k . The k -regularity basically means that the vertices are not distinguished, there is noparticular vertex as, for example, in the case of the star graph, thus we would like to avoidthe cases when the elimination of relatively few vertices would lead to the disintegration ofthe whole comparison system (Tekile, 2017). While the connectedness is really important,because to approximate the decision makers’ preferences well, we need to have at leastindirect comparisons between the different criteria, otherwise we cannot say anything aboutthe relation between certain elements.However, it is also notable that we would like to avoid the cases when two items arecompared only indirectly through a very long path, because this could aggregate the small,tolerable errors of the different comparisons and we could end up with an intolerably largeerror in the relation between the two elements. To measure this problem we can use thediameter of the representing graph: Definition 4 (The diameter of a graph)
The diameter (denoted by d ) of a graph G isthe length of the longest shortest path between any two vertices: d = max u,v ∈ V ( G ) (cid:96) ( u, v ) , here V ( G ) denotes the set of vertices of G and (cid:96) ( ., . ) is the graph distance between twovertices, namely the length of the shortest path between them. Briefly from now on we will examine graphs representing MCDM problems defined bythe following inputs: ( n, k, d ) , where n is the number of vertices (criteria), k shows thelevel of regularity of the graph and d is the diameter of the graph. First of all it is a key step to determine which cases are interesting for us considering ourinputs. It is important to emphasize that we deal with unlabelled graphs, because weare trying to find out what kind of patterns are needed in the comparisons for differentinstances, thus if we exchange the ’names’ of two criteria (like if we would change ’1’ and’2’ in Example 1) the pattern would be the same.Then we can consider the regularity parameter, k . The k = 1 case is possible only when n is even, but they are not connected except for n = 2 , so this is not really interesting forus. When k = 2 there is only one connected graph for every n , namely the cycle, for which d = (cid:98) n/ (cid:99) as already mentioned in the introduction.The larger regularity parameters could be interesting for us, but of course we need areasonable upper bound for the number of criteria, n , which is also an indirect upper boundfor k . In our research we examined the n = 3 , , . . . , cases, because in the one hand forlarger n parameters, some computations become really difficult, and in the other hand wethink that it is reasonable to assume that we do not have more than 20 important criteriaor 20 relatively good alternatives in most of the MCDM problems.The smaller the d parameter is, the more stable or the more trustworthy our systemof comparisons is. This means that in an optimal case we would like to minimize thisparameter, while the number of the criteria ( n ) is always a given exogenous parameterin our MCDM problems. As we mentioned above, k is crucial to avoid the cases whensome criteria (vertices) would be too important in the system, however it also shows ushow many comparisons have to be made, because every vertex has a degree of k , whichmeans the number of edges is nk/ . Thus if our decision makers would like to spend theshortest time with the creation of the PCM, we should choose a small k parameter. But, ofcourse, as usually happens in these situations, there is a trade off between the parameters,6ecause for many criteria (large n ) the smaller regularity ( k ) will cause a bigger diameter( d ), namely a more fragile system of comparisons.In this paper we would like to provide a list of graphs which shows the patterns of thecomparisons that have to be made in case of different parameters. We used an algorithmwhich defines the graph(s) with the smallest diameter ( d parameter) for a given ( n, k ) pair.With the help of these results it was easy to determine which k is the smallest that isneeded to reach a given d for a given n . We found that, with the chosen upper bound of n (20) the interesting values for the regularity are k = 3 , , , while the interesting valuesfor the diameter of the graph are d = 2 , . For a general MCDM problem probably insteadof k , it would give more information if we considered an indicator that shows how far weare from the ’extreme’ case when the decision makers have to make all the comparisons.This would mean n ( n − / comparisons instead of our nk/ in case of regular graphsor ( nk + 1) / in case of quasi-regular graphs, therefore the completion ratio is defined asfollows: c = (cid:40) nk/ n ( n − / if n or k is even ( nk +1) / n ( n − / if n and k are oddthat we will calculate for every instance.Now we will present the results of the algorithm which gives the graphs with the smallestdiameter for a given ( n, k ) pair. Of course d = 1 would mean a complete graph that is notreachable for many ( n, k ) pairs, and also not so interesting for us, thus Table 1 shows thecases when k = 3 and d = 2 is the minimal value of the parameter. It is also importantto note that k = 3 is only possible when n is even, but when it is odd, we examine graphswhere all vertices’ degrees are 3 except one where it is 4, because these are the closest to3-regularity.We can see that with k = 3 the minimal diameter can be 2 until we have 10 vertices. Ofcourse for n ≤ the 3-regularity is not possible, and for n = 4 the diameter is 1, becausethis is a complete graph, but those are really simple (trivial) cases with few possibilities,that is why we skip those in the table. It is also notable that the completion ratio ( c )even reach / when we have 10 vertices (it is obviously decreasing in n ). And we shouldemphasize the fact that there are only a few graphs for every ( n, k ) pair with the minimaldiameter, and one of them is often a bipartite graph that is not the best design in adecision problem, because the two groups are always compared through the other ones(Csató, 2015). Where the table contains ’ ≥ . . . graphs’ that means we have not checked all7 =3 Graph Furtherinformation k =3 Graph Furtherinformation n =5 • 8/10 compar-isons ( c =0 . )• ≥ graphs n =6 c =0 . )• 2 graphs• The othersolution isthe bipartitegraph K , n =7 • 11/21 com-parisons( c ≈ . )• ≥ graphs n =8 Wagnergraph • 12/28 com-parisons( c ≈ . )• 2 graphs n =9 • 14/36 com-parisons( c ≈ . )• ≥ graph n =10 Petersengraph • 15/45 com-parisons( c ≈ . )• Unique graphTable 1: k = 3 -(quasi-)regular graphs on n vertices with minimal diameter d = 2 the possible cases with minimal diameter, but in connection with decision making problemsit is enough to see that there is one pattern that satisfies the needed properties.If we go on to larger graphs ( n > ), then we will find that the smallest reachablediameter changes to d = 3 , but it is also true that at first we have so many graphs thatsatisfies these properties. However as we examine the n = 18 or the n = 20 cases, we cansee that there is only one graph that fulfils our assumptions (Pratt, 1996). The resultsin case of larger graphs, with 3-regularity and 3 as the minimal diameter can be found in8able 2. k =3 Graph Furtherinformation k =3 Graph Furtherinformation n =11 • 17/55 com-parisons( c ≈ . )• ≥ graphs n =12 Tietze graph • 18/66 com-parisons( c ≈ . )• 34 graphs n =13 • 20/78 com-parisons( c ≈ . )• ≥ graphs n =14 Heawood graph • 21/91 com-parisons( c ≈ . )• 34 graphs n =15 • 23/105com-parisons( c = 0 . )• ≥ graphs n =16 • 24/120com-parisons( c = 0 . )• 14 graphs n =17 • 26/136com-parisons( c ≈ . )• ≥ graph n =18 (3,3) graph on18 vertices • 27/153com-parisons( c ≈ . )• Uniquegraph n =19 • 29/171com-parisons( c ≈ . • ≥ graph n =20 (3,3)-graph on 20vertices (C5xF4) • 30/190com-parisons( c ≈ . • Uniquegraph Table 2: k = 3 -(quasi-)regular graphs on n vertices with minimal diameter d = 3 As we can see the completion ratio is still decreasing in n and on larger graphs it can9e taken below 0.2. It is also true that we still do not need to answer for more than 30questions for an MCDM problem with 20 criteria, which can be really useful.We discussed all the possible cases for k = 3 and n ≤ , and we saw that the minimaldiameter is 2 or 3 here. We also mentioned that the 1 diameter would mean a completegraph and a complete PCM, so that is not interesting for us. This means that if we wouldlike to examine the graphs where k = 4 it is obvious that the minimal diameter wouldbe 2 until n = 10 , but it is not so important to make so many comparisons because thisproperty can be reached with k = 3 , too. Thus for k = 4 the interesting cases start above10 vertices, and the question is that can we reach a smaller diameter (a more stable systemof comparisons) with the rise of the answered questions. We found that with k = 4 we canget 2 as the minimal diameter until n = 15 , but for larger values of n , it will be 3 againwhich can be also reached by k = 3 , thus we would not recommend these combinations ofparameters. The results for (11 ≤ n ≤ , k = 4) are shown in Table 3. It is also importantto note that k = 4 is possible in case of both odd and even values of n , thus now we donot have to pay special attention to this.As we can see, the completion ratio is increasing in k , so we cannot get so small c values as in the former table, however the system of comparisons will be more stable evenon many vertices, because the smallest diameter is 2 here. It is also really interestingthat, for larger graphs and regularity levels, the number of connected graphs increase veryrapidly. For instance, when we have 15 vertices, there are 805 491 connected 4-regulargraphs (that means 805 491 possible filling patterns of the PCM), and only one has 2 asits diameter. Our algorithms and methodology has a strong relationship with the so-calleddegree-diameter problem that is well known in the literature of mathematics (Dinneen andHafner (1994), Loz and Sirán (2008)), but they are looking for the largest possible n fora given diameter and a given level of regularity. The scientific results in this field supportour findings, too, because for ( k = 3 , d = 2) the largest n is 10, while for ( k = 3 , d = 3) itis 20. In the case of ( k = 4 , d = 2) the largest n is 15, but for ( k = 4 , d = 3) it is provedthat the largest graph is much above our bound, but the optimal number of the verticesin this case is still an open question.Finally, we can increase the regularity level to 5 in order to find out if we are ableto get 2 as the smallest diameter for larger graphs. The answer is yes, actually it is alsoproven that d = 2 is reachable for 5-regular graphs until 24 vertices, but of course we areinterested in the specific graphs that could help us determine the adequate comparison10 =4 Graph Further information n =11 • 22/55 comparisons ( c =0 . )• 37 graphs n =12 Chvátal graph • 24/66 comparisons ( c ≈ . )• 26 graphs n =13 • 26/78 comparisons ( c ≈ . )• 10 graphs n =14 Unique graph on14 vertices • 28/91 comparisons ( c ≈ . )• Unique graph n =15 Unique graph on15 vertices • 30/105 comparisons( c ≈ . )• Unique graph Table 3: k = 4 -regular graphs on n vertices with minimal diameter d = 2 patterns. Our results can be found in Table 4. The k = 5 parameter is only possible when n is even again, so when it is odd, we let one vertex have 6 as its degree.As we can see in this table there are higher completion ratios again, and for instancewhen we have 20 vertices the decision makers should make 50 comparisons which in certain11 =5 Graph Further information n =16 Clebsch graph • 40/120 comparisons( c ≈ . )• ≥ graphs n =17 • 43/136 comparisons( c ≈ . )• ≥ graph n =18 (18,1)-noncayleytransitive graph • 45/153 comparisons( c ≈ . )• ≥ graph n =19 • 48/171 comparisons( c ≈ . )• ≥ graph n =20 (20,8)-noncayleytransitive graph • 50/190 comparisons( c ≈ . )• ≥ graph Table 4: k = 5 -(quasi-)regular graphs on n vertices with minimal diameter d = 2 situations can be too many. One can also note that in this table we report that there aresome graphs with the needed properties, but never indicate the number of them. Thereason behind this is simple: the really high number of the potential connected 5-regulargraphs (for instance in the case of n = 20 there are roughly · possibilities).This means that we have examined all the cases that we previously called interesting.12ccording to our results if we use the ( n, k, d ) parameters, then for smaller MCDM problemsthe k = 3 is enough to get 2 as the diameter of the representing graph which leads to asmall completion ratio and a stable system of the comparisons. In larger problems, when wehave more alternatives or criteria we can choose if we use k = 3 , when the completion ratiois smaller, but our approximation can be unstable, or choose higher regularity levels withmore reliable results but a higher completion ratio. We also showed examples and graphswith the needed properties for the different cases, which can help anyone in a MCDMproblem to decide which comparisons have to be made. One can find the summary of ourresults in Table 5, which shows how many graphs we know for given ( n, k, d ) parameters.It is also true that if there is a graph for ( n, k, d ) in the table, then there are graphs for ( n, k, D ) too, where D > d , and there is no graph with the parameters of ( n, k, d − . Weomitted the cases when k = 4 and n ≤ , because the minimal diameter is the same as itwas in the case of k = 3 . There is the same reasoning behind the emptiness of the tablewhen k = 5 and n ≤ . We have not included the cases when k = 4 and n ≥ , because d = 3 can be reached by -regular graphs, but for d = 2 at least -regularity is needed. In this article we provided a systematic collection of recommended filling patterns of in-complete pairwise comparisons’ using the graph representation of the PCMs. We discussedthe applied methodology in many details, and then presented our results using the numberof criteria or alternatives, the regularity level and the diameter of the representing graphas parameters. We showed that relatively small completion ratios can be achieved withsmall diameters, and provided examples for every case that we considered to be relevant.The investigation of the robustness of the results, namely what is between the differentregularity levels, could be the topic of a further research. It is also an interesting problemto concentrate directly on the completion ratio as a parameter instead of the regularityof the representing graph. If the ( n, c ) pair is given then what comparisons are the mostimportant to be made? We would like to deal with these questions in our future works.Although our results have been presented within the framework of pairwise comparisonmatrices, they are applicable in a wider range. A lot of other models based on pairwisecomparisons can utilize our findings. For example ranking of sport players or teams basedon their matches leads to the problem of tournament design: which pairs should play13able 5: The summary of the results: the number of k -(quasi-)regular graphs on n nodeswith diameter d against each other? References
Blanquero, R., Carrizosa, E., and Conde, E. (2006). Inferring efficient weights from pairwisecomparison matrices.
Mathematical Methods of Operations Research , 64(2):271–284.Bozóki, S., Csató, L., and Temesi, J. (2016). An application of incomplete pairwise compar-ison matrices for ranking top tennis players.
European Journal of Operational Research ,248(1):211–218. 14sató, L. (2013). Ranking by pairwise comparisons for swiss-system tournaments.
CentralEuropean Journal of Operations Research , 21:783–803.Csató, L. (2015). A graph interpretation of the least squares ranking method.
Social Choiceand Welfare , 44(1):51–69.Dinneen, M. J. and Hafner, P. R. (1994). New results for the degree/diameter problem.
Networks , 24(7):359–367.Gass, S. I. (1998). Tournaments, transitivity and pairwise comparison matrices.
TheJournal of the Operational Research Society , 49(6):616–624.Kendall, M. G. and Smith, B. B. (1940). On the method of paired comparisons.
Biometrika ,31(3/4):324–345.Loz, E. and Sirán, J. (2008). New record graphs in the degree-diameter problem.
Aus-tralasian Journal of Combinatorics , 41:63–80.Pratt, R. W. (1996). The complete catalog of 3-regular, diameter-3 planar graphs.Rezaei, J. (2015). Best-worst multi-criteria decision-making method.
Omega , 53:49–57.Saaty, T. L. (1980).
The Analytic Hierarchy Process . McGraw-Hill, New York.Tekile, H. A. (2017). Incomplete pairwise comparison matrices in multi-criteria decisionmaking and ranking.
MSc Thesis, Central European University . k = 3 and d = 2 graphs Edges1-21-31-41-52-32-4 3-54-515igure 2: n = 5 , k = 3 , d = 2 example’Graph6’ format: D}k1 2 3 4 51 • • • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 6: The bullets in the matrix shows the edges of the graph1 2 3 4 5 61 • • • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ ◦ Table 7: The bullets in the matrix shows the edges of the graphEdges 16igure 3: n = 6 , k = 3 , d = 2 example: 3-prism graph’Graph6’ format: E{Sw1-21-31-42-32-53-64-54-65-6 17igure 4: n = 7 , k = 3 , d = 2 example’Graph6’ format: FsdrO1 2 3 4 5 6 71 • • • • ◦ • • ◦ • • ◦ • • ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ Table 8: The bullets in the matrix shows the edges of the graphEdges1-21-31-41-72-5 2-63-53-64-54-718-7 19igure 5: n = 8 , k = 3 , d = 2 example: Wagner graph’Graph6’ format: GhdHKc1 2 3 4 5 6 7 81 • • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 9: The bullets in the matrix shows the edges of the graphEdges1-21-51-82-3 2-63-43-74-5 4-85-66-77-820igure 6: n = 9 , k = 3 , d = 2 example’Graph6’ format: HsT@PWU1 2 3 4 5 6 7 8 91 • • • ◦ • • ◦ • • • ◦ ◦ • ◦ • • ◦ • • ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ Table 10: The bullets in the matrix shows the edges of the graph21dges1-21-41-52-32-63-43-73-84-95-75-86-86-97-9 22igure 7: n = 10 , k = 3 , d = 2 example: Petersen graph’Graph6’ format: IUYAHCPBG1 2 3 4 5 6 7 8 9 101 • • • • • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 11: The bullets in the matrix shows the edges of the graph23dges1-31-41-62-42-52-73-53-84-95-106-76-107-88-99-10 24
Appendix2: k = 3 and d = 3 graphs Figure 8: n = 11 , k = 3 , d = 3 example’Graph6’ format: J{COXCPAIG_1 2 3 4 5 6 7 8 9 10 111 • • • ◦ • • ◦ ◦ • ◦ • • • ◦ • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 12: The bullets in the matrix shows the edges of the graph25dges1-21-31-42-32-103-74-54-84-115-65-96-76-117-88-99-1010-11 26igure 9: n = 12 , k = 3 , d = 3 example: Tietze graph’Graph6’ format: KhDGHEH_?__R1 2 3 4 5 6 7 8 9 10 11 121 • • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ ◦ ◦ • • ◦ ◦ • ◦ ◦ ◦ Table 13: The bullets in the matrix shows the edges of the graph27dges1-21-91-102-32-63-43-84-54-115-65-96-77-87-128-910-1110-1211-12 28igure 10: n = 13 , k = 3 , d = 3 example’Graph6’ format: LhcIGCP_GGc@_P1 2 3 4 5 6 7 8 9 10 11 12 131 • • • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 14: The bullets in the matrix shows the edges of the graph29dges1-21-51-101-132-32-73-43-124-54-95-66-76-117-88-98-139-1010-1111-1212-13 30igure 11: n = 14 , k = 3 , d = 3 example: Heawood graph’Graph6’ format: MhEGHC@AI?_PC@_G_1 2 3 4 5 6 7 8 9 10 11 12 13 141 • • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 15: The bullets in the matrix shows the edges of the graph31dges1-21-61-142-32-113-43-84-54-135-65-106-77-87-128-99-109-1410-1111-1212-1313-14 32igure 12: n = 15 , k = 3 , d = 3 example’Graph6’ format: N{O___GA?G?k?i?d?J?33 2 3 4 5 6 7 8 9 10 11 12 13 14 151 • • • ◦ • • ◦ ◦ • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 16: The bullets in the matrix shows the edges of the graphEdges1-21-31-42-32-53-64-74-85-95-106-116-12 7-137-148-128-159-129-1310-1410-1511-1311-1512-1434igure 13: n = 16 , k = 3 , d = 3 example’Graph6’ format: O{O___GA?G?_?i?d?K_Ao1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 • • • ◦ • • ◦ ◦ • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 17: The bullets in the matrix shows the edges of the graph35dges1-21-31-42-32-53-64-74-85-95-106-116-127-137-148-158-169-139-1510-1410-1611-1311-1612-1412-15 36igure 14: n = 17 , k = 3 , d = 3 example’Graph6’ format: PhCGKCH?K?_PG@?Cg?GG@c?C1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 • • • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 18: The bullets in the matrix shows the edges of the graph37dges1-21-81-111-172-32-153-43-134-54-175-65-96-76-167-87-128-99-1010-1110-1411-1212-1313-1414-1515-1616-17 38igure 15: n = 18 , k = 3 , d = 3 example: (3,3)-graph on 18 vertices’Graph6’ format: QhCGKCH?G?_PG@?Cg?GG@C?E?GG1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181 • • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ • • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 19: The bullets in the matrix shows the edges of the graph39dges1-21-81-182-32-153-43-134-54-175-65-96-76-167-87-128-99-1010-1110-1411-1211-1812-1313-1414-1515-1616-1717-18 40igure 16: n = 19 , k = 3 , d = 3 example’Graph6’ format: RhECQ?_@G¿@@?C?_G_AO?_S?_G?DG41 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 • • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ • ◦ ◦ ◦ Table 20: The bullets in the matrix shows the edges of the graphEdges1-21-61-72-32-83-43-94-54-145-75-126-10 6-137-178-158-169-109-1810-1111-1211-1612-1513-1413-19424-1715-1916-1717-1818-19 43igure 17: n = 20 , k = 3 , d = 3 example: (3,3)-graph on 20 vertices (C5xF4) ’Graph6’ format: ShECQ?_@G¿@@?C?_G_AO?_??@W@?O?DC
44 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 • • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ • • ◦ ◦ • ◦ ◦ • ◦ ◦ • ◦ • • ◦ ◦ ◦ ◦ ◦ • ◦ ◦ ◦ Table 21: The bullets in the matrix shows the edges of the graphEdges1-21-61-72-32-83-43-94-54-145-75-126-10 6-137-178-158-169-109-1910-1111-1211-1612-1513-1413-20454-1815-2016-1817-1817-1919-20 46
Appendix3: k = 4 and d = 2 graphs Figure 18: n = 11 , k = 4 , d = 2 example: 4-Andrásfai graph’Graph6’ format: JlSggUDOlA_1 2 3 4 5 6 7 8 9 10 111 • • • • ◦ • • • ◦ • • • ◦ ◦ • • ◦ ◦ • • ◦ ◦ • • ◦ ◦ • • ◦ ◦ • • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ Table 22: The bullets in the matrix shows the edges of the graph47dges1-21-41-91-112-32-52-103-43-163-114-54-75-65-86-76-97-87-108-98-119-1010-11 48igure 19: n = 12 , k = 4 , d = 2 example: Chvátal graph’Graph6’ format: KG@LIchdMoV?1 2 3 4 5 6 7 8 9 10 11 121 • • • • • • • • ◦ • • • • • • • • • • • ◦ ◦ • • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 23: The bullets in the matrix shows the edges of the graph49dges1-71-101-111-122-32-62-82-113-73-93-124-84-104-114-125-65-95-115-126-76-107-88-99-10 50igure 20: n = 13 , k = 4 , d = 2 example: 13-cyclotomic graph’Graph6’ format: LhEIHEPQHGaPaP1 2 3 4 5 6 7 8 9 10 11 12 131 • • • • ◦ • • • ◦ • • • ◦ • • • ◦ • • • ◦ ◦ • • ◦ ◦ • • ◦ ◦ • • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ Table 24: The bullets in the matrix shows the edges of the graph51dges1-21-61-91-132-32-72-103-43-83-114-54-94-125-65-105-136-76-117-87-128-98-139-1010-1111-1212-13 52igure 21: n = 14 , k = 4 , d = 2 example’Graph6’ format: Mo?CB‘gXCw@wDgEc?1 2 3 4 5 6 7 8 9 10 11 12 13 141 • • • • ◦ • • • ◦ • • • • • • • • • • • • • • • ◦ • • • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 25: The bullets in the matrix shows the edges of the graph53dges1-21-31-71-112-82-92-103-83-93-104-84-114-134-145-95-115-125-146-106-116-126-137-127-137-148-129-1310-14 54igure 22: n = 15 , k = 4 , d = 2 example’Graph6’ format: N?ACE‘cL?wTGEgQcKP?55 2 3 4 5 6 7 8 9 10 11 12 13 14 151 • • • • • • • • • • • • • • • • • • • • ◦ • • • ◦ • • • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 26: The bullets in the matrix shows the edges of the graphEdges1-61-71-81-122-82-92-142-153-93-103-123-15 4-84-104-114-135-115-125-135-146-96-106-117-1356-147-158-129-1310-1411-15 57
Appendix4: k = 5 and d = 2 graphs Figure 23: n = 16 , k = 5 , d = 2 example: Clebsch graph’Graph6’ format: OPtcIcSoGT@__XWAcJ_ci1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 • • • • • • • • • • ◦ • • • • ◦ • • • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 27: The bullets in the matrix shows the edges of the graph58dges1-31-51-71-101-132-52-62-82-102-143-43-63-143-154-54-84-94-165-115-126-76-96-127-87-117-168-138-159-109-119-13 10-1510-1611-1411-1512-1312-1512-1613-1414-1659igure 24: n = 17 , k = 5 , d = 2 example’Graph6’ format: PxCYHEBCIO_bGPagiAOQP‘@K1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 • • • • • ◦ • • • • ◦ ◦ • • • ◦ • • • • ◦ • • • • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ Table 28: The bullets in the matrix shows the edges of the graph60dges1-21-31-91-141-172-32-72-112-153-43-83-134-54-64-104-155-65-115-145-166-76-126-177-87-97-148-98-138-169-109-14 10-1110-1210-1511-1211-1612-1312-1713-1413-1515-1615-1716-1761igure 25: n = 18 , k = 5 , d = 2 example: (18,1)-noncayley transitive graph’Graph6’ format: Q{eAaSqIWI?o@D@IG?X?WCAkGDo62 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181 • • • • • ◦ • • • • ◦ ◦ • • • ◦ • • • • ◦ ◦ • • • ◦ • • • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 29: The bullets in the matrix shows the edges of the graphEdges1-21-31-41-51-62-32-72-82-153-93-103-16 4-54-74-94-175-85-105-186-116-126-136-147-863-97-128-108-119-109-1410-1311-1411-1611-1712-1312-1612-1813-1513-1714-1514-1815-1715-1816-1716-18 64igure 26: n = 19 , k = 5 , d = 2 example’Graph6’ format: RzAKQQPD@AbOI?O_?Z?IK@BO?rO@FOEdges1-21-31-61-71-92-32-42-82-143-43-113-134-94-104-125-65-7 5-85-125-136-106-166-177-127-147-158-98-158-168-119-189-1910-1110-1565 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 • • • • • ◦ • • • • ◦ ◦ • • • ◦ ◦ • • • • • • • • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ • • • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Table 30: The bullets in the matrix shows the edges of the graph10-1811-1511-1712-1612-1713-1613-1813-1914-1714-1814-1915-1916-1817-19 66igure 27: n = 20 , k = 5 , d = 2 example: (20,8)-noncayley transitive graph ’Graph6’ format: Ssa@Gt‘PQcHOGCGC?cOHAC@cOD_OSgORO
Edges1-21-31-41-51-62-92-102-112-123-73-93-133-144-84-114-174-18 5-85-125-195-206-76-106-156-167-87-117-128-98-109-159-1610-1310-1467 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201 • • • • • ◦ • • • • ◦ • • • • ◦ • • • • ◦ • • • • ◦ • • • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ • • • ◦ ◦ • • • ◦ ◦ ◦ • • ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦ ◦ ◦ ◦ ◦