Graphs Of Valency Research Articles

Half-arc-transitive graphs and chiral hypermaps

A subgroup G of automorphisms of a graph X is said to be 1 2 - arc- transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be 1 2 - arc- transitive if Aut X is 1 2 -arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1 2 -arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1 2 -arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1 2 -arc-transitive group action of the automorphism group G of the original hypermap H . Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤ Aut X acting 1 2 -arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1 2 -arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1 2 -arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1 2 -arc-transitive group action. Indeed, in last section we have applied general results to construct 1 2 -arc-transitive graphs with cycle stabilisers of arbitrarily large orders.

Finite analogues of non-Euclidean spaces and Ramanujan graphs

This is a companion paper of “Finite Euclidean graphs and Ramanujan graphs” by the same authors. Finite analogues of the Poincaré upper half plane, i.e., finite upper half plane graphs, were studied by many researchers, including Terras, Evans etc. Finally, it was proved by combining works of A. Weil, P. Deligne, R. Evans, H. Stark, N. Katz, W. Li and many others, that the finite upper half plane graphs of valency q+1 over the finite field F q are all Ramanujan graphs. In this paper, we obtain further examples of families of Ramanujan graphs, by using previous works on association schemes and the calculations of their character tables, which are in some sense analogues of the finite upper half planes over finite fields, i.e., finite versions of non-Euclidean spaces. A key observation is that in many (but not all) cases, we can obtain a sharper esitimate |θ|≤2 q−2 on eigenvalues, instead of the original |θ|≤2 q , which was proved by Katz. We combine this observation with the ideas of controlling association schemes and the Ennola type dualities, in our previous papers such as Bannai–Hao–Song (J. Combin. Theory Ser. A 54 (1990) 164), Bannai–Hao–Song–Wei (J. Algebra 144 (1991) 189), Bannai–Kwok–Song (Mem. Fac. Sci. Kyushu Univ. Ser. A 44 (1990) 129), Kwok (Graphs Combin. 7 (1991) 39), Tanaka (Master Thesis (2001)), Tanaka (European J. Combin. 23 (2002) 121) and many others. At the end, we remark that for each fixed valency k≥3 there are only finitely many distance-regular Ramanujan graphs of valency k.

Typical circulant double coverings of a circulant graph

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. Kwak and Lee (Canad. J. Math. XLII (1990) 747) enumerated the isomorphism classes of graph bundles and those of n-fold coverings with respect to a group of automorphisms of the base graph G which fix a spanning tree. Hofmeister (Discrete Math. 98 (1991) 175) independently enumerated the isomorphism classes of n-fold graph coverings with respect to the trivial automorphism group of a base graph G. Also, the isomorphism classes of several kinds of graph coverings of a graph G have been enumerated by Hong et al. (Discrete Math. 148 (1996) 85), Hofmeister (Discrete Math. 143 (1995) 87; SIAM J. Discrete Math. 11 (1998) 286), Kwak et al. (SIAM J. Discrete Math. 11 (1998) 273), Kwak and Lee (J. Graph Theory 23 (1996) 105) and some others. In this paper, we aim to enumerate the isomorphism classes of circulant double coverings of a connected circulant graph. The result of our study shows that no double coverings of a circulant graph of valency 3 are circulant. We also enumerate the isomorphism classes of circulant double coverings of a certain type, called a typical covering.

Ten Exceptional Geometries from Trivalent Distance Regular Graphs

The ten distance regular graphs of valency 3 and girth > 4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.

Finite graphs of valency 4 and girth 4 admitting half-transitive group actions

AbstractFinite graphs of valency 4 and girth 4 admitting ½-transitive group actions, that is, vertex- and edge- but not arc-transitive group actions, are investigated. A graph is said to be ½-transitiveif its automorphism group acts ½-transitively. There is a natural orientation of the edge set of a ½-transitive graph induced and preserved by its automorphism group. It is proved that in a finite ½-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to Z2.

Constructing [formula omitted]-arc-transitive graphs of valency 4 and vertex stabilizer [formula omitted

An infinite familiy of 1 2 -arc-transitive graphs of valency 4 with alternating cycles of length 4 is given. Besides, an infinite family of 1 2 -arc-transitive graphs of valency 4 with vertex stabilizer isomorphic to Z 2×Z 2 is constructed.

Fisher's inequality for designs on regular graphs

Abstract Let G=(V,E) be a regular graph of valency d. A (v,k,λ,μ)-design over G is a pair (V, B ) , where B is a family of k-subsets of V (blocks) such that for any distinct vertices x and y, the number of blocks containing {x,y} is equal to λ if {x,y} is an edge and is equal to μ if {x,y} is not an edge. We will prove that the number of vertices does not exceed the number of blocks (Fisher's Inequality) under the following condition: (r−μ)/(μ−λ) is not a multiple eigenvalue of the adjacency matrix of the graph (r is the replication number of the design). We also give examples showing that this restriction is essential.

Arc-Transitive Cayley Graphs of Valency at Most Four on Abelian Groups

In this paper we give a complete classification for arc-transitive and one-regular Cayley graphs of valency at most four on finite abelian groups.

Automorphisms of Aschbacher Graphs

If a regular graph of valence \(k\) and diameter \(d\) has \(v\) vertices, then \(v \leqslant 1{\text{ + }}k{\text{ + }}k\left( {k - 1} \right) + \cdot \cdot \cdot + k\left( {k - 1} \right)^{d - 1} \), which was proved by Moore (cf. [1]). Graphs for which this non-strict inequality turns into an equality are called Moore graphs. Such have an odd girth equal to \(2d + 1\). The simplest example of a Moore graph is furnished by a \(\left( {2d + 1} \right)\)-triangle. Damerell proved that a Moore graph of valence \(k \geqslant 3\) has diameter 2. In this case \(v = k^2 {\text{ + }}1\), the graph is strongly regular with \(\lambda = 0\) and \(\mu = 1\), and the valence \(k\) is equal to 3 (Peterson's graph), to 7 (Hoffman–Singleton's graph), or to 57. The first two graphs are of rank 3. Whether a Moore graph of valence \(k = 57\) exists is not known; yet, Aschbacher proved that the Moore graph with \(k = 57\) will not be a rank 3 graph. We call the Moore graph with \(k = 57\) the Aschbacher graph. Cameron showed that such cannot be vertex transitive. Here, we treat subgraphs of fixed points of Moore graph automorphisms and an automorphism group of the hypothetical Aschbacher graph for the case where that group contains an involution.

Half-transitive graphs of valency 4 with prescribed attachment numbers

Half-transitive graphs of valency 4 with prescribed attachment numbers

A combinatorial construction of a graph with automorphism group [formula omitted

A simple combinatorial construction is given which takes as its imput a regular graph of valency k such that the convex closure of two points at distance two is the complete bipartite graph K 3,3 and whose output is a regular graph of valency 2k+1. It is shown that the sequence of graphs obtained by starting with the graph with one point and no edges and applying the construction recursively is the family of bipartite dual polar space of type DSO +(2n,2) .

Distance-Regular Graphs of Valency 6 and a1 = 1

We give a complete classification of distance-regular graphs of valency 6 and a1 = 1.

Constructing Infinite One-regular Graphs

A graph is said to be one-regular if its automorphism group acts regularly on the set of its arcs. A construction of an infinite family of infinite one-regular graphs of valency 4 is given. These graphs are Cayley graphs of almost abelian groups and hence of polynomial growth.

The Distance-Regular Graphs of Valency Four

We show that each distance-regular graph of valency four has known parameters.

Constructing 4-Valent 12-Transitive Graphs with a Nonsolvable Automorphism Group

A graph is said to be 1 2 -transitive if its automorphism group acts transitively on vertices and edges but not on arcs. For each n ⩾11, a 1 2 -transitive graph of valency 4 and girth 6, with the automorphism group isomorphic to A n × Z 2 , is given.

Half-Transitive Group Actions on Finite Graphs of Valency 4

The action of a subgroupGof automorphisms of a graphXis said to be 12-transitiveif it is vertex- and edge- but not arc-transitive. In this case the graphXis said to be (G, 12)-transitive. In particular,Xis 12-transitiveif it is (Aut X, 12)-transitive. The 12-transitive action ofGonXinduces an orientation of the edges ofXwhich is preserved byG. LetXhave valency 4. An even length cycleCinXis aG-alternating cycleif every other vertex ofCis the head and every other vertex ofCis the tail of its two incident edges in the above orientation. It transpires that allG-alternating cycles inXhave the same length and form a decomposition of the edge set ofX(Proposition 2.4); half of this length is denoted byrG(X) and is called theG-radiusofX. Moreover, it is shown that any two adjacentG-alternating cycles ofXintersect in the same number of vertices and that this number, called theG-attachment number aG(X) ofX, divides 2rG(X) (Proposition 2.6). IfXis 12-transitive, we let theradiusand theattachment numberofXbe, respectively, the Aut X-radius and the Aut X-attachment number ofX. The caseaG(X)=2rG(X) corresponds to the graphXconsisting of twoG-alternating cycles with the same vertex sets and leads to an arc-transitive circulant graph (Proposition 2.4). IfaG(X)=rG(X) we say that the graphXistightly G-attached. In particular, a 12-transitive graphXof valency 4 istightly attachedif it is tightly Aut X-attached. A complete classification of tightly attached 12-transitive graphs with odd radius and valency 4 is obtained (Theorem 3.4).

Maps and Half-transitive Graphs of Valency 4

A subgroupG of automorphisms of a graphX is said to be 12- transitive if it is vertex- and edge- but not arc-transitive. The graphX is said to be 12-transitive if AutX is 12-transitive. The correspondence between regular maps and 12-transitive group actions on graphs of valency 4 is studied via the well known concept of medial graphs. Among others it is proved that under certain general conditions imposed on a map, its medial graph must be a 12-transitive graph of valency 4 and, vice versa, under certain conditions imposed on the vertex stabilizer, a 12-transitive graph of valency 4 gives rise to\\break an irreflexible regular map. This way infinite families of 12-transitive graphs are constructed from known examples of regular maps. Conversely, known constructions of 12-transitive graphs of valency 4 give rise to new examples of irreflexible regular maps. In the end, the concept of a symmetric genus of a 12-transitive graph of valency 4 is introduced. In particular, 12-transitive graphs of valency 4 and small symmetric genuses are discussed.

A ½‐transitive graph of valency 4 with a nonsolvable group of automorphisms

A ½‐transitive graph of valency 4 with a nonsolvable group of automorphisms

A �-transitive graph of valency 4 with a nonsolvable group of automorphisms

A graph X is said to be ½-transitive if its automorphism group acts transively on the sets of its vertices and edges but intransitively on the set of its arcs. A construction of a ½-transitive graph of valency 4 and girth 6 with a nonsolvable group of automorphism is given. © 1997 John Wiley & Sons, Inc. J Graph Theory 25, 1: 133–138, 1997

A Family of One-regular Graphs of Valency 4

A graph is said to beone-regularif its automorphism group acts regularly on the set of its arcs. A construction of an infinite family of one-regular graphs of valency 4 with vertex stabilizerZ22having a non-solvable group of automorphisms is given. The smallest graph in this family has 60 vertices.