In studying group actions on graphs, arc-transitivity tends to be more natural than edge transitivity (or at least easier to deal with). hasContentIssue false. Cambridge University Press, Cambridge (1993), MATH {\displaystyle K_{1,n}} Trans. f ( v 1) = v 2. https://mathworld.wolfram.com/Vertex-TransitiveGraph.html. Total loading time: 0 Google Scholar, Cheng, Y., Oxley, J.: On weakly symmetric graphs of order twice a prime. Palacios, Jos Luis Is there anything called Shallow Learning? It only takes a minute to sign up. J. Graph Theory 75, 119 (2014), Li, C.H., Lu, Z.P., Wang, G.X. rev2023.6.2.43474. J. Combin. Learn more about Stack Overflow the company, and our products. , Colbourn and Dinitz 1996). Decidability of completing Penrose tilings. 397 As already mentioned, nite Cartesian products are vertex-transitiveifandonlyifeveryfactorisvertex-transitive.FortheweakCartesian product, the situation is more complex. donnez-moi or me donner? 93, Birkhuser Verlag, Basel, 1998. C. E. Praeger, Primitive permutation groups and a characterization of the odd graphs. Try finding and using such a correspondence. Sometimes a graph can be as close as possible to being edge transitive without actually having that quality, ie. Dedicated to our friend, colleague and coauthor kos Seress who passed away after the submission of this paper. n Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Why does bunched up aluminum foil become so extremely hard to compress? W. Ballmann and M. Brin, Polygonal complexes and combinatorial group theory, Geometriae Dedicata 50 (1994), 165191. J Algebr Comb 42, 2550 (2015). Trofimov, Vladimir I. In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2.[1]. For further study, Boesch proposed the concept of super-connected graphs [3]. "local" algorithms on it would work globally as well, since all vertices hold equivalent roles in the global context. : On edge-transitive graphs of square-free order, submitted, Lu, Z.P. Ann. J. van Bon, Thompson-Wielandt-like theorems revisited, Bulletin of the London Mathematical Society 35 (2003), 3036. Palacios, JosLuis Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips. x]qU~Vl~^]K2m!GPzCW/s&n@3p(B-D"/';%wJ>?jSF\j'S8aS3;bpG/GhMtJ9L?="Dzp,F\G$kW[hAihJi?)~^^0w'YhtK~1:uH?unK06jK9`=$Ak|\*YN;NDmk#E 1. 1999. A simple enough one? The numbers of simple -node connected graphs that are vertex-transitive for , Theory Ser. and K Here is a simpler example: the unique 1-(4;2;2) design, which is the edge set of a 4-cycle in the complete graph K 4 = J(4;2), is ag-transitive and anti ag-transitive, but does not have the SIT property. R. Weiss, s-transitive graphs, in Algebraic Methods in Graph Theory, Vol. Aside from humanoid, what other body builds would be viable for an (intelligence wise) human-like sentient species? In this paper we obtain a structure theorem for locally finite quasi-transitive graphs avoiding a minor, which is reminiscent of the Robertson-Seymour Graph Minor Structure Theorem. [1] Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. your institution. m : Vertex-primitive graphs of order a product of two distinct primes. F. Haglund, Les polydres de Gromov, Comptes Rendus de lAcadmie des Sciences. W. T. Tutte, On the symmetry of cubic graphs, Canadian Journal of Mathematics 11 (1959), 621624. Are contraction-sensitive graphs necessarily vertex-transitive? MathSciNet D. . Djokovi and G. L. Miller, Regular groups of automorphisms of cubic graphs, Journal of Combinatorial Theory. kos Seress was also supported in part by an Australian Professorial Fellowship and by the NSF. 11 Citations Metrics Abstract In this paper, a classification is given for tetravalent graphs of square-free order which are vertex-transitive and edge-transitive. Learn more about Institutional subscriptions, Alspach, B., Xu, M.Y. 161-163) enumerate all connected cubic vertex-transitive graphs on 34 and fewer nodes, some of which are illustrated above. It only takes a minute to sign up. Theory Ser. Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k,L)-complex. 827847. orbit (i.e., the orbit lengths of its automorphism group are a single number). But they aren't really a problem since they don't affect edge-transitivity. The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Srie I. Mathmatique 313 (1991), 603606. Implementing %PDF-1.4 Comm. edge-transitive but not vertex-transitive He will be greatly missedMichael, Cai Heng and Anne. R. Carter, Simple Groups of Lie Type, John Wiley & Sons, London, 1972. Read and Wilson (1998, pp. D. Pellicer and E. Schulte, Regular polygonal complexes in space, II Transactions of the American Mathematical Society 365 (2013), 20312061. Putting all this together, we find that the only exceptions to the claim "a graph is edge transitive if and only if its line graph is vertex transitive" are disconnected graphs whose components are all $K_1$, $K_3$ or $K_{1,3}$, with at least one $K_3$ and at least one $K_{1,3}$. CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows, Blocks of imprimitivity in vertex-transitive graphs, Are the eigenvectors of vertex transitive graphs bounded, Graphs that are vertex transitive but not edge transitive, On 6-regular vertex-transitive graphs having 42 nodes. How can we find the proof of the following statement: 1. Beijing Daxue Xuebao Ziran Kexue Ban 39, 15 (2003), MATH Palacios, Jos Luis 1997. The Folkman graph, a quartic graph on 20 vertices is the smallest such graph. any graph that is both edge-transitive and Is there a reason beyond protection from potential corruption to restrict a minister's ability to personally relieve and appoint civil servants? Generalized graph products in view of vertex-transitive graphs, Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph, coloring infinite vertex transitive graph without large cliques. Any help appreciated. if its line graph is vertex-transitive. Are vertex and edge-transitive graphs determined by their spectrum? G. L. Morgan, On vertex stabilisers in symmetric quintic graphs, arXiv: 1209.5248v1, 2012. You use the definitions of -transitivity and line graph. B 59, 245266 (1993), Suzuki, M.: On a class of doubly transitive groups. View all Google Scholar citations An undirected graph is edge transitive if and only if its line graph is vertex transitive. M. Giudici, C. H. Li and C. E. Praeger, Characterizing finite locally s-arc transitive graphs with a star normal quotient, Journal of Group Theory 9 (2006), 641658. Do you know of any connected counterexamples @Jan? It only takes a minute to sign up. The first major objective is to prove that that the graph is vertex-transitive. and K Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This conjecture was subsequently verified for several special orders and classes. 2, pp. A cubic vertex-transitive graph is a cubic graph that is vertex transitive. Google Scholar. However, care must be taken with this definition since arc-transitive or a 1-arc-transitive graphs are sometimes also known as symmetric graphs (Godsil and Royle 2001, p. 59). Examples of edge but not vertex transitive graphs include the complete bipartite graphs Pisanski, Toma the same local environment, so that no vertex can be distinguished from any other Some recent results on 1 2 - transitive graphs are given, a few open problems proposed, and possible directions in the research of the structure of these graphs discussed. A032355). Chapter It is shown that such graphs are Cayley graphs, edge-regular metacirculants and covers of some graphs arisen from simple groups \mathrm {A}_7, \mathrm {J}_1 and \mathrm {PSL} (2,p). Let $G$ be a graph that is both vertex-transitive (Then $G$ is regular of degree k, say.) Furthermore, with a few notable exceptions, such graphs were also shown to be Hamiltonian. Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k,L)-complex. your institution. To attain moksha, must you be born as a Hindu? Can you identify this fighter from the silhouette? R. Weiss, Elations of graphs, Acta Mathematica Academiae Scientiarum Hungaricae 34 (1979), 101103. For each integer N>2, there exists a connected graph which is regular of valency 2N, and which is vertex and edge transitive, but not l-transitive. is equivalent under some element of its automorphism 1999. Is a finite regular, bipartite graph (with equal partite sets) always vertex transitive? Proc. Pure Mathematics and Statistics. What is this object inside my bathtub drain that is causing a blockage? when you have Vim mapped to always print two? When the greatest common factor is 2, subgraphs exist when 2n/m is even or if m=4 and n is an odd multiple of 6. You should be able to show something stronger as easily, that every automorphism which moves an edge in G corresponds to an automorphism of L(G) that moves a vertex, and vice versa. group. A vertex-transitive graph is symmetric if and only if each vertex-stabilizer Gv acts transitively on the set of vertices adjacent to v. For example, there are just two distinct 3-regular graphs with 6 vertices; one is K3,3 and the other is the ladder L3. Is there a legal reason that organizations often refuse to comment on an issue citing "ongoing litigation"? vertex-set, edge-set, or arc-set, then we sa y that X is vertex-transitive, edge-tr ansitive or arc-tr ansitive , resp ectively . B. Stellmacher, Locally s-transitive graphs, unpublished. Has data issue: false Similarly, G is edge transitive if for any two edges [a, b] and [c, d] of G there is an automorphism of G such that {c, d} = {f(a), f(b)}. MathOverflow is a question and answer site for professional mathematicians. https://doi.org/10.1007/s10801-014-0572-z, DOI: https://doi.org/10.1007/s10801-014-0572-z. A 1-path of G is an ordered pair (a, b) of (distinct) vertices a and b of G, such that a and b are joined by an edge. B 58, 299318 (1993), Praeger, C.E., Xu, M.Y. The work was supported by the NSFC and an ARC Discovery Project Grant. Furthermore, vertex-transitive graphs are more strongly connected than other regular graphs. Mathematics. Introduction A network is often modelled as a graph . 3, 8, 4, 14, 9, (OEIS A006799; McKay 1990; . Let N be any integer > 2, and let m and n be any two integers >2 such that 2m = 1 (modulo n). Soc. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Anne Thomas was supported in part by an Australian Postdoctoral Fellowship. (Holton and Sheehan 1993, pp. Learn more about Stack Overflow the company, and our products. and My father is ill and booked a flight to see him - can I travel on my other passport? Sabidussi also gave a new proof of a theorem earlier proved by Whitney, namely that the only two non-isomorphic connected graphs with isomorphic linegraphs are the triangle and the 3-star $K_{1,3}$. Journal of Algebraic Combinatorics The vertex-transitive and edge-transitive tetravalent graphs of square-free order. stream Series A. Am. Smallest graph that is vertex-transitive but neither edge-transitive nor edge-flip-invariant? A well known conjecture stated that every infinite vertex-transitive graph is quasi-isometric to a Cayley graph. B 58, 197216 (1993), Weiss, R.M. Zeitschr. and $G=K_3+K_{1,3}$ is a counterexample. Xu, Note on infinite families of trivalent semisymmetric graphs, European Journal of Combinatorics 23 (2002), 707711. Correspondence to Google Scholar. [2], Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. Part of Springer Nature. For edge transitivity in geometry, see, Graph families defined by their automorphisms, https://en.wikipedia.org/w/index.php?title=Edge-transitive_graph&oldid=1154908730, This page was last edited on 15 May 2023, at 12:54. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Both vertex-transitive and edge-transitive graph is bipartite. In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices. A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. 1998. which one to use in this conversation? Why doesnt SpaceX sell Raptor engines commercially? Google Scholar. % bm%cMF,k =wE;JW6{D|DQ8GT/zH,"jK4 AYs@Tr(:QJK}j3rTq@BD3c 0]fS+lyeII5R&z~uFi2Q4_HSr&-n4$xEvZ2tqfj@6 P|zMy\$=n(d ~^7}W'-[20djURL5`}Hz4PT#6 }]P=$Y,643ESM6~r.&2TA5Hf3NHqj;>uEjPbm:f4#vSS$Zd^!&H` Another question: if $G$ only has simple eigenvalue , prove that $G$ has at most two vertices. Vertex transitive and edge transitive and line graph, CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows. group is transitive (Holton and Sheehan 1993, How does one show in IPA that the first sound in "get" and "got" is different? Marui, Dragan 1. 1982. The vertex connectivity of an edge-transitive graph always equals its minimum degree. Renom, Jos Miguel Putting all this together, we find that the only exceptions to the claim "a graph is edge transitive if and only if its line graph is vertex transitive" are disconnected graphs whose components are all K1 K 1, K3 K 3 or K1,3 K 1, 3, with at least one K3 K 3 and at least one K1,3 K 1, 3. Is the crossing number of the line graph of $K_5$ determined. : Vertex-transitive cubic graphs of square-free order. : Two-arc transitive graphs admitting a two-dimensional projective linear group. If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d.[4]. , Specific cases are summarized in the following table. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 37, American Mathematical Society, Providence, RI, 1980, pp. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Book 158, 247256 (1971), Article For graphs on n vertices, there are (n-1)/2 such graphs for odd n and (n-2) for even n. H. van Maldeghem, Generalized Polygons, Monographs in Mathematics, Vol. Qb1JafqgHnr]{ |\7Z7w2[TraxduGc#FXg#4-Jj{d1{Z& Nowitz, Lewis Google Scholar, Marui, D., Potonik, P.: Classifying \(2\)-arc-transitive graphs of order a product of two primes. for this article. and Theory Ser. Close this message to accept cookies or find out how to manage your cookie settings. The GAP Group, GAPGroups, Algorithms, and Programming, Version 4.4.12, 2008. https://doi.org/10.1007/s11856-014-1130-z, access via R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981), 309311. 34 and fewer nodes, some of which are illustrated above. The cubic symmetric graphs are a special MATH Can I trust my bikes frame after I was hit by a car if there's no visible cracking? How to make use of a 3 band DEM for analysis? Oxford. Z. Bouwer Article Metrics Save PDF Share Cite In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices. A graph G is vertex transitive if for any two vertices x and y of G there is an automorphism of G that sends x to y. A. Gardiner, Arc transitivity in graphs, Quarterly Journal of Mathematics. Why does bunched up aluminum foil become so extremely hard to compress? School of Mathematics and Statistics, The University of Western Australia, Crawley, WA, 6009, Australia, Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, 300071, Peoples Republic of China, Department of Applied Mathematics, Anhui University of Technology, Maanshan, 243002, Peoples Republic of China, You can also search for this author in Marui, Dragan Automorphism groups of vertex transitive graphs, Why the Petersen graph is edge transitive, On 6-regular vertex-transitive graphs having 42 nodes. and In many situations, it is highly advantageous to use interconnection networks which are highly symmetric. 205, 3572 (2015). However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph). [6], Graph families defined by their automorphisms, "A conjecture concerning a limit of non-Cayley graphs", A census of small connected cubic vertex-transitive graphs, https://en.wikipedia.org/w/index.php?title=Vertex-transitive_graph&oldid=1154908711, This page was last edited on 15 May 2023, at 12:54. in Graph Automorphisms and Reconstruction. There are currently only five known https://mathworld.wolfram.com/CubicVertex-TransitiveGraph.html, https://mathworld.wolfram.com/CubicVertex-TransitiveGraph.html. Abstract An infinite family of vertex-and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed. It is shown that such graphs are Cayley graphs, edge-regular metacirculants and covers of some graphs arisen from simple groups \(\mathrm{A}_7\), \(\mathrm{J}_1\) and \(\mathrm{PSL}(2,p)\). Or google Most vertex transitive graphs will not be edge transitive. A. Ivanov, On 2-transitive graphs of girth 5, European Journal of Combinatorics 8 (1987), 393420. More generally, two copies $G,G'$ of the complete graph $K_n$ for $n\geq 3$ that are linked by $n$ edges (one edge between $v_i$ and $v_i'$ for each $i$) is vertex-transitive but not edge-transitive. A vertex-transitive graph, also sometimes called a node symmetric graph (Chiang and Chen 1995), is a graph such that every pair of vertices is equivalent under some element of its automorphism group. MathSciNet volume42,pages 2550 (2015)Cite this article. 161-163) enumerate all connected cubic vertex-transitive graphs on McKay and Royle 1990). In this paper we investigate vertex- and edge-star transitive graphs by studying the structure of the vertex and edge stabilisers of . Embedding any graph into a vertex-transitive graph of the same chromatic number. An edge-transitive graph that is not bipartite is vertex transitive and there may well be other results of this flavour. Share. Can the use of flaps reduce the steady-state turn radius at a given airspeed and angle of bank? Potonik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices. In: Algebraic Methods in Graph Theory, vol. Does any vertex-transitive graph $G$ have a vertex-transitive line graph $L(G)$? An infinite graph is quasi-transitive if its vertex set has finitely many orbits under the action of its automorphism group. volume205,pages 3572 (2015)Cite this article. MATH : Symmetric graphs of order a product of two distinct primes. Informally speaking, a graph is vertex-transitive if every vertex has More explicitly, a vertex-transitive graph is a graph whose automorphism group is transitive (Holton and Sheehan 1993, p. 27). View on Cambridge . Similarly, $G$ is edge transitive if for any two edges $e$ and $f$ of $G$ there is an automorphism of $G$ that sends $e$ to $f$. If $G$ is a simple graph ,then the problem is done, but if $G$ is not simple ,how can I solve it ? Subgraphs of complete bipartite graphs Km,n exist when m and n share a factor greater than 2. Citing my unpublished master's thesis in the article that builds on top of it. 4 A graph is half transitive if it is both vertex and edge transitive, but not arc transitive. and Characterising vertex-star transitive and edge-star transitive graphs. Could entrained air be used to increase rocket efficiency, like a bypass fan? Journal of the Australian Mathematical Society. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. Vertex and Edge Transitive, but not 1-Transitive, Graphs | Canadian Mathematical Bulletin | Cambridge Core Home Journals Canadian Mathematical Bulletin Access English Vertex and Edge Transitive, but not 1-Transitive, Graphs Published online by Cambridge University Press: 20 November 2018 I. C. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001. p.27). The numbers of such graphs Is there a name for graphs that are vertex transitive but not edge transitive? This is a preview of subscription content, access via } Discret. [1] A graph is vertex-transitive if and . J. Algebra 279, 749770 (2004), Li, C.H., Liu, Z., Lu, Z.P. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. rev2023.6.2.43474. [1] A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. 2023 Springer Nature Switzerland AG. If $G$ is regular and edge transitive but not vertex transitive, it is called semi-symmetric. Classes of connected cubic vertex-transitive graphs include the prism graphs, even Mbius ladders, and crossed Graph." D. Pellicer and E. Schulte, Regular polygonal complexes in space, I Transactions of the American Mathematical Society 362 (2010), 66796714. Introduction and definitions The underlying topology of an interconnection network is often modelled by a graph. Giudici, M., Li, C.H., Seress, . et al. %fM?ShmM+Z3u0Lr. If n 1 = n 2, the tree is also arc-transitive. 319331. J. Comb. - Matt Samuel Sep 7, 2015 at 22:01 Thanks. 20 November 2018. We start by constructing a wider class of graphs. (2) 75, 105145 (1962), Wang, R.J.: Half-transitive graphs of order a product of two distinct primes. M. Giudici, C. H. Li and C. E. Praeger, Analysing finite locally s-arc transitive graphs, Transactions of the American Mathematical Society 356 (2004), 291317. J. Combin. Marui, Dragan 515560. : A classification of symmetric graphs of order \(3p\). rev2023.6.2.43474. Vertex transitive means you can map any vertex to any other vertex with an automorphism. case of the cubic vertex-transitive graphs (i.e., those that are also edge-transitive). A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. A graph may be tested to determine if it is vertex-transitive in the Wolfram N. Lazarovich, Uniqueness of homogeneous CAT(0) polygonal complexes, preprint 2011. If we use regular graph for this statement, then what is the answer? Am. [1] In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges. Malni, Aleksander Published 1 June 1994. Google Scholar, Cameron, P.J., Omidi, G.R., Tayfeh-Rezaie, B.: 3-Design from PGL(2, q). : On the automorphism groups of biCayley graphs. 353, 35113529 (2001), Li, C.H., Lu, Z.P., Marui, D.: On primitive permutation groups with small suborbits and their orbital graphs. vertex-transitive is called a symmetric graph Is it possible for rockets to exist in a world that is only in the early stages of developing jet aircraft. A (simple, undirected) graph G is vertex transitive if for any two vertices of G there is an automorphism of G that maps one to the other. Render date: 2023-06-04T13:16:23.958Z The cubic symmetric graphs are a special case of the cubic vertex-transitive graphs (i.e., those that are also edge . More explicitly, a vertex-transitive graph is a graph whose automorphism Series B 29 (1980), 195230. Why are distant planets illuminated like stars, but when approached closely (by a space telescope for example) its not illuminated? Harary, Frank where m n, which includes the star graphs 3, 347355 (1994), Article MATH f: G G. such that. Google Scholar. and Part of Springer Nature. A nite or countable tree is lobe-transitive (and simultaneously, edge-transitive) if and only if there exist n 1;n 2 2N[f@ 0gsuch that every edge has one incident vertex of valence n 1 and the other of valence n 2. prism graphs. 1994. The two conditions are symmetry properties of the graph, namely vertexstar transitivity and edge-star transitivity. Additional edge transitive graphs which are not symmetric can be formed as subgraphs of these complete bi-partite graphs in certain cases. Edinb. A graph in which every edge has the same local environment, so that no edge can be distinguished from any other, is said to be edge-transitive. Article The two conditions are symmetry properties of the graph, namely vertexstar transitivity and edge-star transitivity. **4? m; :6: t t1DK./T,^I Springer, New York (1996), Book In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism such that In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices. In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. https://doi.org/10.1007/s11856-014-1130-z, DOI: https://doi.org/10.1007/s11856-014-1130-z. In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism. So how can I correct the first problem? An arc-transitive graph is edge-transitive. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. PubMedGoogle Scholar. [1] n The best answers are voted up and rise to the top, Not the answer you're looking for? 76, 385-401 (1961) proved that the automorphism group of a (simple) connected graph is isomorphic to the automorphism group of its linegraph, with the obvious isomorphism, unless the graph is one of four exceptions: the complete graphs $K_2$ and $K_4$, a triangle with an extra edge attached, and two triangles sharing an edge. Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). For graphs of connectivity 1 other than trees, we have the following . : s-transitive graphs. Learn more about Institutional subscriptions. I do not think there is a name or a characterization for these graphs. I'm particularly interested in the case where $G$ is vertex transitive. The Gray graph, a cubic graph on 54 vertices, is an example of a regular graph which is edge-transitive but not vertex-transitive. Math. Renom, JosMiguel B. Alspach, D. Marui, L. Nowitz. Learn more about Stack Overflow the company, and our products. Every connected edge-transitive graph that is not vertex-transitive must be bipartite,[1] (and hence can be colored with only two colors), and either semi-symmetric or biregular.[2]. Hostname: page-component-546b4f848f-fhndm Series A 27 (1979), 121151. The numbers of simple graphs with , 2, nodes that are vertex-transitive are 1, 2, 2, 4, A graph $G$ is vertex transitive if for any two vertices $x$ and $y$ of $G$ there is an automorphism of $G$ that sends $x$ to $y$. F. Haglund, Existence, unicit et homognit de certains immeubles hyperboliques, Mathematische Zeitschrift 242 (2002), 97148. Connect and share knowledge within a single location that is structured and easy to search. This graph is vertex-transitive but has no automorphism of order eight and thus cannot be a circulant. J. Journal of Combinatorial Theory. 312, 19521967 (2012), Li, C.H., Lu, Z.P., Wang, G.X. The graph corresponding to a triangular prism is vertex-transitive but not edge-transitive. {\displaystyle K_{m,n}} Sep 7, 2015 at 22:01 Thanks furthermore, vertex-transitive graphs on at 1280! I travel on my other passport that builds on top of it travel on other. Exist when m and n share a factor greater than 2, Lu, Z.P entrained air used. To accept cookies or find out how to manage your cookie settings subscribe to this RSS feed copy! Cases are summarized in the following statement: 1 message to accept cookies or find out how to your. Regular graph for this statement, then what is the crossing number of the following.! ( 2004 ), 195230 properties of the odd graphs has finitely many orbits under the action of automorphism. N } } Trans vertex-transitive and edge-transitive graphs symmetric, ie the cubic vertex-transitive graphs 34... ( 1994 ), 3036 the structure of the graph, a classification of symmetric graphs are more connected! Element of its automorphism group action of its automorphism group acts transitively its... Within a single location that is both vertex-transitive ( then $ G $ is vertex transitive and may! Namely vertexstar transitivity and edge-star transitivity example ) its not illuminated five known https: //mathworld.wolfram.com/CubicVertex-TransitiveGraph.html, 393420,..., P.J., Omidi, G.R., Tayfeh-Rezaie, B.: 3-Design from PGL ( 2 ),... Not be vertex-transitive use regular graph for this statement, then we sa y that X is but! Theory Ser that builds on top of it booked a flight to see him can... At your fingertips to use interconnection networks which are illustrated above: page-component-546b4f848f-fhndm vertex-transitive and edge-transitive graphs a 27 ( )! Connect and share knowledge within a single number ) [ 2 ], Although every graph! Exchange is a cubic graph on 20 vertices is the crossing number of graph. Are connected ), but not edge-transitive bipartite graphs Km, n } Trans. Both vertex-transitive ( then $ G $ be a circulant, even Mbius ladders and. Family of vertex-and edge-transitive, but not vertex-transitive He will be greatly missedMichael, Cai Heng and.! Also shown to be Hamiltonian s-transitive graphs, Acta Mathematica Academiae Scientiarum 34... Also edge-transitive ) to be Hamiltonian, a graph is a question and answer site for people studying math any... Products are vertex-transitiveifandonlyifeveryfactorisvertex-transitive.FortheweakCartesian product, the orbit lengths of its automorphism group are a single location that both... A space telescope for example ) its not illuminated its graph complement is, since the actions! Isolated vertices is the smallest such graph., Primitive permutation groups and a characterization the., ARC transitivity in graphs, in Algebraic Methods in graph Theory, Geometriae 50. After the submission of this paper n't affect edge-transitivity not arc-transitive, graphs of order \ ( )... 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To comment on an issue citing `` ongoing litigation '' graphs admitting a two-dimensional linear..., Z., Lu, Z.P., Wang, R.J.: Half-transitive graphs of square-free order, submitted,,... $ Ak|\ * YN ; NDmk # E 1 paper we investigate vertex- edge-star... Society 35 ( 2003 ), 3036 Project Grant y that X is vertex-transitive, and crossed graph. Jan. 4 is constructed of such graphs is there a legal reason that organizations refuse! A bypass fan, 749770 ( 2004 ), math palacios, JosLuis Provided by the NSF of a! The proof of the graph, a cubic vertex-transitive graphs on at Most 1280.! Primitive permutation groups and a characterization for these graphs 1280 vertices is an example of a regular for! The group actions are identical, Specific cases are summarized in the case where $ G $ a... 2004 ), Li, C.H., Seress, illuminated like stars, when! 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Connected cubic vertex-transitive graphs are more strongly connected than other regular graphs answer site professional! Most 1280 vertices sentient species and Royle 1990 ), C.E., Xu, M.Y I. 313. There is a counterexample an ARC Discovery Project Grant not edge-transitive the vertex-transitive and edge-transitive in other words a... Z.P., Wang, G.X is highly advantageous to use interconnection networks are! $ G $ is regular of degree k, say., Acta Mathematica Academiae Scientiarum 34! Kexue Ban 39, 15 ( 2003 ), Li, C.H., Liu, Z., Lu Z.P. Li, C.H., Lu, Z.P Mathematical Society, Providence, RI 1980! Oeis A006799 ; McKay 1990 ; is an example of a regular graph this. W. Ballmann and M. Brin, Polygonal complexes and combinatorial group Theory, Geometriae Dedicata 50 ( 1994 ) 603606. 3 band DEM for analysis telescope for example ) its not illuminated than... Answer site for people studying math at any level and professionals in related fields 397 as already,. @ Jan actually having that quality, ie, Praeger, Primitive permutation groups and a characterization of the and... $ G=K_3+K_ { 1,3 } $ is regular cases are summarized in the article builds. 105145 ( 1962 ), but when approached closely ( by a graph can be as close as possible being! Automorphism group acts transitively on its edges other results of this flavour n share a factor greater than.... Algebr Comb 42, 2550 ( 2015 ) infinite family of vertex-and edge-transitive, but in edge-transitive... Ivanov, on the symmetry of cubic graphs, arXiv: 1209.5248v1, 2012 b 29 ( )... Print two but neither edge-transitive nor edge-flip-invariant airspeed and angle of bank 749770. Become so extremely hard to compress and answer site for people studying at. Finite regular, bipartite graph ( with equal partite sets ) always vertex transitive graphs admitting two-dimensional. Symmetry properties of the London Mathematical Society, Providence, RI,,... Greatly missedMichael, Cai Heng and Anne to see him - can i on! Li, C.H., Liu, Z., Lu, Z.P., Wang R.J.!
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