G. Sabidussi, "Vertex-transitive graphs," Monatsheft fur Mathematik, vol. 68, pp. 426--438, 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as a Schreier coset graph, generalising the representation of a Cayley graph above: we replace group elements by cosets of a subgroup H of G as vertices of the graph, and for adjacency, in place of an inverse closed set of elements we use an inverseclosed set of double cosets of H in G. Sabidussi [46] used this representation to show that any vertex transitive graph has a multiple which is a Cayley graph. Here, a multiple of G is obtained by replacing each vertex by a coclique of size k, and each edge by all possible edges between the corresponding cocliques, for some k. Not every ....

G. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426{ 438.

....of Gnfidg, the Cayley graph of G relative to S, denoted by Cay(G; S) has vertex set G and edges of the form fg; gsg, for all g 2 G and s 2 S. By the definition, the group G acting by right multiplication is a subgroup of Aut Gamma and acts regularly on V Gamma = G. The converse also holds (see [6]) A circulant is a Cayley graph of a cyclic group. Thus a graph Gamma is a circulant of order n if and only if Aut Gamma contains a cyclic subgroup of order n which is regular on V Gamma. A classification of 2 arc transitive circulants was given in [1] It was proved that a connected, ....

G. O. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964).

....subgroup of G and that the quotient group G=G 0 is abelian [13, Thms. 3.4.11 and 3.4.10, p. 59] Since G=G 0 is abelian and transitive on V (X=G 0 ) it follows from the next result that X=G 0 is a Cayley graph on the abelian group G= G x G 0 ) for any x 2 V (X) Lemma 2. 10 (Sabidussi [12]) If G x is trivial for some x 2 V (X) then X is (isomorphic to) a Cayley graph on G. 3 Preliminaries on the Frattini subgroup As in Section 2, we assume that Assumption 2.1 holds. Assumption 3.1. We assume G 0 is cyclic of order p k , where p is a prime, and that X has at least three ....

G. Sabidussi, Vertex-transitive graphs, Monatshefte fur Math., 68 (1964) 426-- 438.

....of at most O(n) Using Stirling s approximation completes the proof. 1 A graph is vertex transitive if Aut[ Gamma] acts transitively on V , i.e. if for any two vertices x and y there is an automorphism a such that y = a(x) For instance, all Cayley graphs have this property, see e.g. [47]. 11 Garc ia Pelayo Stadler: The XY Hamiltonian Theorem 3. The discrete XY Hamiltonian is elementary on C n ff with eigenvalue = ae 4 if ff = 2 8 sin 2 ( ff) if ff 3 for all n 2 and all ff 2. The correlation length is = n 4 sin 2 ( ff) Proof. A simple calculation ....

G. Sabidussi. Vertex transitive graphs. Mh. Math, 68:426--438, 1964.

....only fx; yg is an edge. The set of all automorphisms forms the automorphism group Aut[ Gamma] A graph is vertex transitive if Aut[ Gamma] acts transitively on V , i.e. if for any two vertices x and y there is an automorphism a such that y = a(x) All Cayley graphs have this property, see e.g. [33]. The action of Aut[ Gamma] on the set of all ordered pairs of vertices induces a partition R of V Theta V the classes of which are the orbits Aut[ Gamma] The class X containing a particular pair (x 0 ; y 0 ) of vertices is X = Phi Gamma a(x 0 ) a(y 0 ) Delta j a 2 Aut[ Gamma] Psi . ....

G. Sabidussi. Vertex transitive graphs. Mh. Math, 68:426--438, 1964.

.... GnH. The vertex set is given by the set of left cosets fgHjg 2 Gg and there is an arc (g 1 H; g 2 H) whenever g 1 sH = g 2 H for some s 2 S. Cayley coset digraphs are jSj regular, connected and vertex symmetric. Moreover, every vertex symmetric digraph is a Cayley coset digraph, as is shown in [13]. In particular, a Cayley coset digraph Gamma is a Cayley digraph iff H = feg, where e is the identity of G. Gamma = Cay(G; H;S) is a hierarchical or quasi minimal Cayley coset digraph iff there is an ordering of the elements of S, say fs 1 ; s 2 ; Delta Delta Delta ; s k g such that for ....

G. Sabidussi, Vertex transitive graphs, Monatsh. Math. 68 (1969) 426--438.

....configuration induced on V Theta V . Then: G is transitive ( G is homogeneous G is multiplicity free ( G is commutative G is generously transitive ( G is symmetric Proof. See [84] The first condition is of particular importance: All Cayley graphs have transitive groups of automorphisms [85], and thus the corresponding coherent configurations G are homogeneous. The Hamming graphs and the Johnson graphs, for example, are distance transitive, i.e. for any two pairs of vertices (u; v) and (x; y) fulfilling d(x; y) d(u; v) there is an automorphism ff such that ff(x; y) u; v) ....

G. Sabidussi. Vertex transitive graphs. Mh. Math, 68:426--438, 1964.

....H 0 ; Phi) with vertex set V = H and edge set f(x; y)jxy Gamma1 2 Phi is called the Cayley graph of the group H with respect to the set of generators Phi. Not all vertex transitive graphs are Cayley graphs, however. A famous counter example is the Petersen graph, figure 2. Sabidussi [106] gives a method for constructing all vertex transitive graphs from groups which is closely related to Cayley graphs. Let be an equivalence relation on the vertex set of a graph Gamma. Then the graph Gamma= has the equivalence classes as its vertices, and two such vertices are joined by an ....

....such that Gamma = Gamma(H; H 0 ; Phi) 2) A graph Gamma is a Cayley graph if and only if its automorphism group contains a subgroup G which acts regularly on the vertex set. Then Gamma = Gamma(G; Phi) with an appropriate set of generators. Proof. The first statement is due to Sabidussi [106], the second one is proved in [11, 12] Lateron we will need a few properties of simple random walks on vertex transitive graphs. If Gamma is vertex transitive, let us define the probability s that a simple random walk of s steps ends in a vertex contained in the symmetry class . An explicit ....

G. Sabidussi. Vertex transitive graphs. Mh. Math, 68:426--438, 1964.

....(Cayley coset digraphs, to be precise) and that of sequences over an alphabet can be used as the underlying representation, and each has its own advantage. The former idea was the point of departure for the discovery of the cycle prefix network, motivated by the fundamental theorem of Sabidussi [11] which shows that all vertex symmetric digraphs are Cayley coset digraphs. The sequence representation of a cycle prefix network is more useful for studying its properties and for implementing it in practice. For this reason, we shall utilize the sequence representation of the cycle prefix digraph ....

G. Sabidussi, Vertex transitive graphs, Monatsh. Math. 68 (1969), 426-438.

....under the above assumptions, the compatibility condition implies that the voltage assignment ff is completely determined by the distribution of voltages on the arcs emanating from the vertex id 2 . Clearly, combining Corollary 2 with the well known (directed version of) theorem of Sabidussi [18], we can conclude that the lift G ff is necessarily a Cayley graph as well. However, we shall be interested in the structure of the underlying group of the lift. For that reason we recall the concept of semidirect product Theta OE Gamma of the groups and Gamma (which depends on the above ....

G. Sabidussi, Vertex transitive graphs, Monatsh. Math. 68 (1969) 426-438.

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G. Sabidussi, "Vertex-transitive graphs," Monatsheft fur Mathematik, vol. 68, pp. 426--438, 1964.

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Sabidussi, G.: Vertex--transitive graphs. Monatsh. Math. 68 (1964) 426--438

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G. Sabidussi. Vertex--transitive graphs. Monatsh. Math., 68:426--438, 1964.

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Sabidussi, G. O., Vertex-transitive graphs, Monatshefte fur Math. 68 1964, 426-438.

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G. Sabidussi, "Vertex-transitive graphs," Monatsheft fur Mathematik, vol. 68, pp. 426--438, 1964.

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Sabidussi, G. O., Vertex-transitive graphs, Monatshefte fur Math. 68 1964, 426-438.

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G. Sabidussi. Vertex-transitive graphs. Monatsh. Math. 68:426-438, 1964.

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G. Sabidussi, #Vertex transitive graphs," Monatsh. Math., 68 #1969#, 426#438.

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-183. #13# G. Sabidussi. Vertex transitive graphs, Monatshefte f#ur Mathematik

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