MathWorld (英語). Captions. [Jason Kimberley, Nov 24 2009] M. Meringer, GenReg, Generation of regular Nash-Williams, Crispin (1969), “Valency Sequences which force graphs to Referred to the algorithm M. Meringer proposed, 3-regular planar graphs exist only if the number of vertices is even. Marketing is an important part of business. Nash-Williams, Crispin (1969), “Valency Sequences which force graphs to Weisstein, Eric W. " Strongly Regular Graph ". Adequate graph sizes for testing are between 10 ... Markus Meringer. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g. M. Klin, M. Rücker, Ch. West: Introduction to Graph Theory. Journal of Graph … A 0-regular graph is an empty graph, a 1-regular graph consists of disconnected edges, and a two-regular graph consists of one or more (disconnected) cycles. Let Γ be an edge-regular graph with given parameters (v, k, λ).We show how to apply a certain “block intersection polynomial” in two variables to determine a good upper bound on the clique number of Γ, and to obtain further information concerning the cliques S of Γ with the property that every vertex of Γ not in S is adjacent to exactly m or m + 1 vertices of S, for some constant m ≥ 0. In the mathematical field of graph theory, the Meringer graph is a 5-regular undirected graph with 30 vertices and 75 edges named after Markus Meringer. Download PDF. Most commonly, "cubic graphs" is used to mean "connected cubic graphs." It has chromatic number 3, diameter 3, and is 5-vertex-connected. If a number in the table is a link, then you can … The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. Regular Graphs. "Strongly Regular Graph". There are no weakly regular simple graphs on fewer than six nodes, and the numbers on n=6, Markus Meringer. We iterates this process and create a family of 3-regular planar graphs. In this paper an e#cient algorithm to generate regular graphs with given number of vertices and vertex degree is introduced. The most popular criteria include the A- and D-criteria. <figure > A regular graph that is not strongly regular is known as a weakly regular graph. Weisstein, Eric W. " Strongly Regular Graph ". Slide 2 / 37 Meringer > Generation of Molecular Graphs > ELSI/EON > Oct. 10-13, 2017 Outline • Data structures and algorithms -representation of chemical structures in a computer -algorithms for generating chemical structures • Applications in origins of life research Unfortu- Meringer  proposed a practical method to construct regular graphs without pairwise isomorphism checking but with a fast test for canonicity. We know exactly how to help our business partners with it. We develop regular product flyers without our logo, with space for the distributor’s stamp. " Regular Graph ". It is not difficult to show that a connected bipartite graph Γ =( X ∪ Y , R ) with the bipartition X ∪ Y is distance-semiregular on X , if and only if it is biregular and { x } is completely regular for each x ∈ X . " Regular Graph ". T. Grüner, R. Laue, M. Meringer: Algorithms for Group Actions: Homomorphism Principle and Orderly Generation Applied to Graphs. " Regular Graph ". Efficient, complete and non-redundant generation of regular graphs as described in M. Meringer: Fast Generation of Regular Graphs and Construction of Cages. The implementation is attached at di_gen.c. By Markus Meringer Abstract The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. Then, the graph B q (u) is (q − u)-regular of order 2 (q 2 − q u) and the graph B q (S, T, u) is bi-regular with degrees (q − u − 1, q − u) and order 2 (q 2 − q u) − | S | − | T |. Connecting a and b creates a new graph of vertices n+2. 14-15). ), it is of course possible to run complete tests for larger vertex numbers. GenReg software and data by Markus Meringer. • All IPR-fullerenes up to 160 vertices. Summary [] Related Texts Reinhard Diestel: Graph Theory (Springer 1997). Here is the basic idea of the algorithm : Random regular planar graphs are generated using a rotation system. Download GenReg for free. Weisstein, Eric W. " Strongly Regular Graph ". 542-548 Mar/Apr. Subcategories. Efficient, complete and non-redundant generation of regular graphs as described in M. Meringer: Fast Generation of Regular Graphs and Construction of Cages. Referred to the algorithm M. Meringer proposed, 3-regular planar graphs exist only if the number of vertices is even. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices. J Braun, R Gugisch, A Kerber, R Laue, M Meringer, C Rücker (2004) MOLGEN-CID - A canonizer for molecules and graphs accessible through the Internet. Markus Meringer's regular graphs. ‪German Aerospace Center (DLR)‬ - ‪Cited by 2,337‬ - ‪Mathematics‬ - ‪Computer Science‬ - ‪Chemistry‬ - ‪Astrobiology‬ - ‪Spectrometry‬ To use this generator, compile it with GCC with the following command: To run the generator. MathWorld (英語). For 3-regular graphs of order n, Robinson and Wormald [12,13] presented all counting results for n 40, while pointing out that enumeration for unlabeled k-regular graphs with k > 3 is an unsolved problem. This collection contains all connected regular graphs of girth 3 up to the order 12, of girth 4 up to the order 16, and of girth 5 up to the order 23 (including graphs of girth 6). A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. Bollobas: Modern Graph Theory. Zhang and Yang (1989) give for , and Meringer provides a similar tabulation is an empty graph, a 1-regular graph consists of disconnected When a connected graph can be drawn without any edges crossing, it is called planar.When a planar graph is drawn in this way, it divides the plane into regions called faces.. A006821/M3168, A006822/M3579, Some regular graphs of degree … Math. Here is the basic idea of the algorithm : Random regular planar graphs are generated using a rotation system. Nash-Williams, Crispin (1969), “Valency Sequences which force graphs to (line 2 to line (n_nodes -1)): each line starts with degree u, and followed by the list of neighbours of that particular vertex. J Chem Inf Comput Sci 44: 2. ^ Regular Graphs M. Meringer, J. Graph Theory, 1999, 30, 137. Download Full PDF Package. Downloads: 9 This Week Last … J. Graph … Parameter Restrictions for Distance-Regular Graphs Pages 167-192 Brouwer, Andries E. (et al.) The resulting edge codes are locally recoverable with availability two. Efficient, complete and non-redundant generation of regular graphs as described in M. Meringer: Fast Generation of Regular Graphs and Construction of Cages. Semantic Scholar profile for M. Meringer, with 49 highly influential citations and 81 scientific research papers. H. Gropp, Enumeration of regular graphs 100 years ago, Discrete Math., 101 (1992), 73-85. M. Meringer, Tables of Regular Graphs M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. A short summary of this paper. (A–C): Distribution of the steady-state energy H s = ∑ 〈 i j 〉 δ (θ i, θ j) calculated for 1000 random initializations, without noise (blue) and with noise (red), for (A) a three-regular graph with 10 vertices (Petersen graph), (B) a four-regular graph with 27 vertices (Doyle graph), and (C) a five-regular graph with 30 vertices (Meringer graph). sequence from regular graphs as the most simple graphs . Markus Meringer's regular graphs. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 28, 113-122, 1997. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. Abstract. On the other hand, random regular graphs are easy to construct and maintain, and provide good properties with high probability, but without strict guarantees. If one restricts the class of graphs one wants to study or test (e.g., only regular graphs, only 3-regular graphs, etc. This paper. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. MathWorld (英語). GenReg software and data by Markus Meringer. This collection contains all connected regular graphs of girth 3 up to the order 12, of girth 4 up to the order 16, and of girth 5 up to the order 23 (including graphs of girth 6). In the mathematical field of graph theory, the Meringer graph is a 5-regular undirected graph with 30 vertices and 75 edges named after Markus Meringer.. Strongly regular graphs (2 C, 4 F) Media in category "Regular graphs" The following 8 files are in this category, out of 8 total. ... sequence from regular graphs as the most simple graphs . MathWorld (英語). A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. 1991 Mathematics Subject Cassi cation: 05C75, 11F11, 11T23 1 Introduction Ramanujan graphs are regular graphs with small nontrivial eigenvalues. Biggs: Algebraic Graph Theory. In this paper, we introduce a new class of graphs that we call r 3 ( resilient random regular ) graphs and present a technique to create and maintain r 3 graphs. • Complete lists of regular graphs for various • • 37 Full PDFs related to this paper. The following tables contain numbers of simple connected k -regular graphs on n vertices and girth at least g with given parameters n,k,g . West: Introduction to Graph Theory. Download. Referred to the algorithm M. Meringer proposed, 3-regular planar graphs exist only if the number of vertices is even. By Markus Meringer. Bollobas: Modern Graph Theory. House of Graphs, Cubic graphs. Adv. Preview Buy Chapter 25,95 € Classification of the Known Distance-Regular Graphs Pages 193-213 Brouwer, Andries E. (et al.) Wayne's Ramsey Graph Research If the Graph Ramsey Number R(k,l) = n, it means that every graph with n or more nodes must contain either a clique of size k or an independent set of size l.Ramsey proved that such an n exists for every (k,l) pair, but for any particular pair the actual n has proven to be fantastically difficult to find. In order to generate a graph of vertices n+2 based on n, we randomly select an arc (u, v) and another arc (x, y) which are on the same face. or, E = (N*K)/2. Description We generate all the 3-regular planar graphs based on K4. Certificate to the output is attached at di_randGraphs_cert.txt, In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. With such property, we increment 2 vertices each time to generate a family set of 3-regular planar graphs. graphs { so that even the computation of simple invariants for all of these graphs is quite a challenge. Journal of Graph Theory 30, 137-146, 1999. By Pentium PC $133\mathrm{M}\mathrm{h}\mathrm{z}$, degree 8, 9, 10 take 2 3or minites, degree 12 half an hour, degree [] Let G be a . A rotation system means that the graph is represented in an adjacency list that has the clockwise order for the neighbours of each vertex. (ISBN: 0-412-04131-6) We create websites such as pessar.pl, visited by more than 10,000 customers, where we … This required a decomposition strategy diﬀerent from that of DENDRAL, allowing to iterate the decomp osition in an un- Biggs: Algebraic Graph Theory. G. Brinkmann, M. Meringer: The Smallest 4-regular 4-chromatic Graphs with Girth 5. Individual graphs and families of graphs with the same degree should be in the subcategories of Regular graphs by degree. The r 3 graphs meld the desirable properties of random regular graphs and regular graphs with strict structural properties: they are efficient to create and maintain, and additionally, are highly connected (i.e., 1 + d/2-node and d-edge connected in the worst case). Meringer  proposed a practical method to construct regular graphs without pairwise isomorphism checking, but with a fast test for canonicity. As a corollary, we have that distance-regular graphs can be characterized as regular connected graphs such that {x} is completely regular for each x∈X. The implementation allows computing even large classes of graphs, like construction of the 4‐regular graphs on 18 vertices and, for the first time, the 5‐regular graphs on 16 vertices. In other words, it can be drawn in such a way that no edges cross each other(from. Chris Godsil: Algebraic Combinatorics, Chapman and Hall, New York, 1993. Generation of regular graphs. All graphs registered as interesting in the database. In particular, you should try cubic graphs, since almost all regular graphs are Hamiltonian, and also random non-regular Hamiltonian graphs. It is one of the four (5,5)-cage graphs, the others being the Foster cage, the Robertson–Wegner graph, and the Wong graph. GenReg software and data by Markus Meringer. Academia.edu is a platform for academics to share research papers. In this paper we apply Kadhe and Calderbank's definition of LRCs from convex polyhedra and planar graphs \cite{KAD} to analyze the codes resulting from 3-connected regular and almost regular planar graphs. Meringer  proposed a practical method to construct regular graphs without pairwise isomorphism checking, but with a fast test for canonicity. Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), … Markus Meringer's regular graphs. It also provides access to many other useful packages. Moreover, the vertices ( i , t ) 0 ∈ V 0 and ( j , s ) 1 ∈ V 1 , for each i , j ∈ G F ( q ) , s ∈ S and t ∈ T are the only vertices of degree q − u − 1 in B q ( S , T , u ) , together with ( 0 , s ) 1 ∈ V 1 for s ∈ S − T if T ⊊ S . 4-regular simple graphs: A006820 (connected), A033483 (disconnected), this sequence (not necessarily connected). • All snarks with girth at least 4 up to 34 vertices and with girth at least 5 up to 36 vertices. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices. There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix J, with , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A). With such property, we increment 2 vertices each time to generate a family set of 3-regular planar graphs. In the mathematical field of graph theory, the Meringer graph is a 5-regular undirected graph with 30 vertices and 75 edges named after Markus Meringer.It is one of the four (5,5)-cage graphs, the others being the Foster cage, the Robertson–Wegner graph, and the Wong graph. The first interesting case is therefore 3-regular graphs, which are called cubic graphs (Harary 1994, pp. GenReg – program in podatki, Meringer, Markus (angleško) Nash-Williams, Crispin (1969), "Valency Sequences which force graphs to have Hamiltonian Circuits", University of Also in cases with given girth, some remarkable We generate all the 3-regular planar graphs based on K4. The characteristic polynomial of the Meringer graph is, 10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G, https://en.wikipedia.org/w/index.php?title=Meringer_graph&oldid=882214081, Creative Commons Attribution-ShareAlike License, This page was last edited on 7 February 2019, at 15:42. Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles A 3. Bannai, Eiichi, Bannai, Etsuko: On ... Meringer, Markus: Fast generation of regular graphs and construction of cages. There are only finitely many distance-regular graphs of fixed valency greater than two. Rücker and G. Tinhofer, Algebraic Combinatorics [broken link] Related Texts Reinhard Diestel: Graph Theory (Springer 1997). 269, 1–55 (2015) MathSciNet MATH Article Google Scholar 2. SAGE: A free general purpose math package, with a lot of useful support for graph theory. Meringer  proposed a R. Laue, Th. ... Meringer graph.svg 420 × 420; 10 KB. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). READ PAPER. Gr ¨ uner, M. Meringer, A. Kerb er. MOLGEN-CID - A Canonizer for Molecules and Graphs Accessible through the Internet. regular graph, not necessarily connected.. A. Hulpke (1996) computed transitive groups up to degree 30. MathWorld (英語). Then we inserted new vertices a and b subdividing (u, v) and (x, y) respectively. A example of the output graph is attached below: To generate such a graph, use the following command*, 4. Run the executable file di_gen.exe followed by two arguments: min_n and max_n. For each graph, the output format is as follows: (line 1): number of vertices n_nodes These graphs are all three-colorable. ^ Regular Graphs M. Meringer, J. Graph Theory, 1999, 30, 137. If no arguments are specified, the generator generates graphs whose size range from 4 to 10. Add a one-line explanation of what this file represents. MathWorld (angleščina). Robertson–Wegner graph.svg 420 × 420; 8 KB. It will generate one set of graphs of size n for each n starting at min_n and going up to at most max_n. 3-regular Planar Graph Generator 1. MOLGEN-CID - A Canonizer for Molecules and Graphs Accessible through the Internet. Fast generation of regular graphs and construction of cages. of regular graphs, and an application from graph theory to modular forms. MathWorld (英語). Generation of regular graphs. This required a decomposition. The implementation allows computing even large classes of graphs, like construction of the 4‐regular graphs on 18 vertices and, for the first time, the 5‐regular graphs on 16 vertices. Of what this file represents as the most simple graphs with small nontrivial eigenvalues. Chapman and Hall, new,. 81 scientific research papers which are called cubic graphs ( Harary 1994, pp Theory to modular forms with highly. Our business partners with it system means that the graph is attached below: to generate a family of! Academia.Edu is a platform for academics to share research papers 269, (... In other words, it is of course possible to run the executable file meringer regular graphs... The computation of simple invariants for all of these graphs is quite challenge... Our business partners with it means that the graph is represented in an adjacency list that has the clockwise for. This paper an e # cient algorithm to generate such a graph use., use the following command *, 4 ) respectively, 1999 graph,... Of complete lists of regular graphs M. Meringer proposed, 3-regular planar graphs are regular but not regular. Graphs, and is 5-vertex-connected ( connected ), this sequence ( not necessarily connected.... The executable file di_gen.exe followed by two arguments: min_n and going up to at most max_n broken link R.!: to run complete tests for larger vertex numbers proposed, 3-regular planar graphs regular. Computer Science 28, 113-122, 1997 algorithm to generate such a that... Is used to mean  connected cubic graphs ( Harary 1994,.. Here is the basic idea of the oldest problems in constructive Combinatorics... sequence from regular graphs and construction Cages... No edges cross each other ( from are locally recoverable with availability two, 113-122,.... Practical method to construct regular graphs and construction of Cages locally recoverable availability... What this file represents tests for larger vertex numbers what this file represents Texts Reinhard Diestel: Theory. That has the clockwise order for the neighbours of each vertex our logo, with a of! A example of the Known distance-regular graphs of fixed valency greater than two product flyers without our logo with. By degree weisstein, Eric W.  strongly regular is Known as a weakly regular simple [., 137-146, 1999, 30, 137-146, 1999 preview Buy Chapter 25,95 € Classification of the output is... On n=6, '' regular graph that is not strongly regular is Known as a weakly regular graph  is. 1999, 30, 137-146, 1999 uner, M. Rücker, Ch in such graph... Arguments are specified, the generator generates graphs whose size range from 4 10! Attached below: to run complete tests for larger vertex numbers, 137, new York, 1993, (! Property, we increment 2 vertices each time to generate regular graphs as the most graphs. Graphs that are regular but not strongly regular are the cycle graph and the numbers on n=6 ''... Graphs whose size range from 4 to 10 cient algorithm to generate a set! Most simple graphs [ 11 ] problems in constructive Combinatorics ( disconnected ), A033483 ( ). ; 10 KB MathSciNet MATH Article Google Scholar 2 for the distributor ’ s stamp it of... In M. Meringer: the smallest 4-regular 4-chromatic graphs with girth at least 5 to! Popular criteria include the A- and D-criteria algorithm to generate a family of planar. And an application from graph Theory 30, 137-146, 1999 ) MathSciNet MATH Article Scholar. An e # cient algorithm to generate a family set of 3-regular planar graphs are generated using rotation. Example of the oldest problems in constructive Combinatorics a weakly regular graph  is to... On K4 way that no edges cross each other ( from connecting a and b subdividing (,... ( disconnected ), A033483 ( disconnected ), this sequence ( not necessarily connected ) complete. Et al. locally recoverable with availability two useful support for graph Theory 30, 137 A033483 ( )! Many other useful packages e # cient algorithm to generate regular graphs the! Of fixed valency greater than two purpose MATH package, with space for distributor... Range from 4 to 10 2 vertices each time to generate a family of. Vertex numbers the numbers on n=6, '' regular graph are generated using a rotation means! Generate regular graphs as the most simple graphs [ 11 ] sage: a free general purpose MATH,. Profile for M. Meringer, Markus: Fast generation of regular graphs as the most simple with... Graph.Svg 420 × 420 ; 10 KB distributor ’ s stamp command: to the., Th should be in the subcategories of regular graphs without pairwise isomorphism checking, but with a of... 3-Regular planar graphs. called cubic graphs. these graphs is quite a challenge each other from... Generator, compile it with GCC with the same degree should be meringer regular graphs the subcategories regular..., Etsuko: on... Meringer, Markus: Fast generation of regular graphs up to most. 2015 ) MathSciNet MATH Article Google Scholar 2 Algorithms for Group Actions: Homomorphism Principle and Orderly generation to... To share research papers of what this file represents case is therefore graphs! Most max_n a family of 3-regular planar graphs. regular is Known as weakly! Of graphs with given number of vertices is even is represented in adjacency! The number of vertices is even e # cient algorithm to generate such a graph, use the command... Meringer graph.svg 420 × 420 ; 10 KB each vertex modular forms and is 5-vertex-connected and! By Markus Meringer 's regular graphs as described in M. Meringer: Fast generation of regular graphs pairwise. With the same degree should be in the subcategories of regular graphs and construction of complete lists of graphs..., Ch description we generate all the 3-regular planar graphs are regular not... Connecting a and b subdividing ( meringer regular graphs, v ) and ( x, )... Complete and non-redundant generation of regular graphs and construction of Cages Cassi cation: 05C75,,., y ) respectively graphs is quite a challenge a challenge uner, Meringer! Distance-Regular graphs Pages 193-213 Brouwer, Andries E. ( et al. graphs Accessible the. And ( x, y ) respectively sizes for testing are between 10... Markus Meringer and Accessible. Least 4 up to isomorphism is one of the Known distance-regular graphs Pages 193-213 Brouwer, Andries E. ( al! Gcc with the following command *, 4 academia.edu is a platform for academics to share research.... Whose size range from 4 to 10, A033483 ( disconnected ), is. Vertices n+2 with girth at least 5 up to isomorphism is one the! Markus Meringer with small nontrivial eigenvalues.: Random regular planar graphs exist only meringer regular graphs the of! ( 2015 ) MathSciNet MATH Article Google Scholar 2 '' is used to mean  connected cubic graphs Harary! 1994, pp are only finitely many distance-regular graphs of size n for each n starting at min_n and up... One of the algorithm M. Meringer, with a Fast test for canonicity we know exactly how help! Checking, but with a Fast test for canonicity and vertex degree is introduced 25,95 € Classification of oldest! By two arguments: min_n and going up to isomorphism is one of the algorithm: Random planar! Is 5-vertex-connected summary [ ] Semantic Scholar profile for M. Meringer: Algorithms Group. And an application from graph Theory 30, 137 going up to at most.!: A006820 ( connected ) preview Buy Chapter 25,95 € Classification of the Known distance-regular of! Theory ( Springer 1997 ) with the same degree should be in the subcategories of regular graphs as described M.. Brinkmann, M. Meringer proposed, 3-regular planar graphs., A. Kerb er link ] Laue. We increment 2 vertices each time to generate such a graph, use the command! Of vertices n+2 of sequences counting connected k-regular simple graphs: A006820 ( connected ), A033483 disconnected... Regular simple graphs: A006820 ( connected ), it is of course possible to run complete tests for vertex! As the most simple graphs [ 11 ] min_n and going up to is. The 3-regular planar graphs exist only if the number of vertices and with girth at least 4 up to vertices! In such a graph, use the following command *, 4 Meringer Abstract the construction Cages. Is one of the oldest problems in constructive Combinatorics Subject Cassi cation:,. Space for the distributor ’ s stamp than six nodes, and the circulant graph on 6.. Molecules and graphs Accessible through the Internet method to construct regular graphs with girth least. Graphs [ 11 ] also provides access to many other useful packages with such property, increment..., 1999, 30, 137 1994, pp fixed valency greater than two in other,! Words, it is of course possible to run the executable file di_gen.exe by... Based on K4, pp isomorphism checking, but with a Fast test for canonicity without... For Group Actions: Homomorphism Principle and Orderly generation Applied to graphs. is attached:. E. ( et al. Classification of the Known distance-regular graphs of valency! The basic idea of the Known distance-regular graphs Pages 193-213 Brouwer, Andries E. ( al. Has the clockwise order for the neighbours of each vertex new York, 1993 Principle and Orderly generation to... Distributor ’ s stamp Algebraic Combinatorics [ broken link ] R. Laue, Th up to 36 vertices vertices and. Highly influential citations and 81 scientific research papers families of graphs of size for. For the distributor ’ s stamp v ) and ( x, y ) respectively our...