Vertex-transitive Graphs Research Articles

Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method

We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.

D-Levels locally homogeneous graphs

This article introduces the concept of a D-levels locally homogeneous graph which forms a special type of self-centered graphs. Some results relating D-levels locally homogeneous graphs to diametrical graphs (which are also self-centered) are obtained. Diametrical graphs of small order (≤ 8) are shown to be D-levels locally homogeneous. When the diameter is 2, it is shown that the concept of a D-levels locally homogeneous graph and that of a diametrical graph coincide. Bipartite diametrical graphs of diameter 3 and regular S-graphs of diameter 4 are characterized as D-levels locally homogeneous graphs. We also prove that the vertex-transitive graphs are precisely the totally D-levels locally homogeneous graphs.

On Small Separations in Cayley and Vertex Transitive Graphs

AbstractDeVos and Mohar proved a rough structure theorem about small separations in vertex‐transitive graphs [5, 6]. By using a new version of Varopolous isoperimetric inequality, we give an improvement of their lower bound on the expansion in the case of infinite vertex‐transitive graphs. Specifically, let be a locally finite connected graph such that there is a group G acting discretely and transitively on X. If is nonempty and finite such that is connected and , then X has a ring‐like structure. Moreover, we give a similar asymptotic result under the assumption that X is an infinite vertex transitive graph. In the setting of finite groups, we use local expansion to show the existence of a nontrivial cyclic subgroup with an effectively bounded index. Finally, we prove that for any there is a graph T and a subgraph A of T, such that .

ON ISOMORPHISMS OF VERTEX-TRANSITIVE CUBIC GRAPHS

We study the isomorphism problem of vertex-transitive cubic graphs which have a transitive simple group of automorphisms.

Multi-Way Expanders and Imprimitive Group Actions on Graphs

For |$n\geq 2$|⁠, the concept of |$n$|-way expanders was defined by many researchers. Bigger |$n$| gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all |$n$| for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on the graph.

On the distribution of distances in homogeneous compact metric spaces

We provide a simple proof that in any homogeneous, compact metric space of diameter D, if one finds the average distance A achieved in X with respect to some isometry invariant Borel probability measure, thenD2≤A≤D. This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds.

Vertex-transitive graphs that have no Hamilton decomposition

It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.

Linking rings structures and semisymmetric graphs: Cayley constructions

An LR structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR structures were introduced in Potočnik and Wilson (2014) as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we consider algebraic methods of constructing LR structures, using number theory, Cayley graphs, affine groups, abelian groups and fields.

On Embeddings of Circulant Graphs

A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).

Matching preclusion for cube-connected cycles

Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. The matching preclusion number of a graph G with even order is the minimum number of edges whose deletion results in a graph without perfect matchings and the conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without perfect matchings. We consider matching preclusion of cube-connected cycles network CCCn. By using the super-edge-connectivity of vertex-transitive graphs, the super cyclically edge-connectivity of CCCn for n=3,4 and 5, Hall’s Theorem and the strengthened Tutte’s Theorem, we obtain the matching preclusion number and the conditional matching preclusion number of CCCn and classify respective optimal matching preclusion sets.

A Note on Coloring Vertex-Transitive Graphs

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\}$, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin–Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.

Generalized Cages

Let $2 \leq k_1 < k_2 < \ldots < k_t $, $3 \leq g_1 < g_2 < \ldots < g_s < N$ be integer parameters. A $(k_1,k_2,\ldots,k_t;g_1,g_2,\dots,g_s;N)$-graph is a graph that contains vertices of degrees $k_1,k_2,\ldots,k_t$ but no other degrees and cycles of lengths $g_1,g_2,\dots,g_s$ but no other cycles of length $< N$. For any given set of parameters satisfying the above conditions, we present an explicit construction of $(k_1,k_2,\ldots,k_t;g_1,g_2,\dots,g_s;N)$-graphs and extend the concept of a cage (a smallest graph of given degree and girth) to that of a generalized cage -- a smallest $(k_1,k_2,\ldots,k_t;g_1,g_2,\dots,g_s;N)$-graph. We introduce several infinite families of generalized cages and study their basic properties in the context of connected, bipartite, and vertex-transitive graphs, as well as combinatorial configurations (in the context of multilaterals).

LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

AbstractWe define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

On automorphism groups of graph truncations

It is well known that the Petersen graph, the Coxeter graph, as well as the graphs obtained from these two graphs by replacing each vertex with a triangle, are trivalent vertex-transitive graphs without Hamilton cycles, and are indeed the only known connected vertex-transitive graphs of valency at least two without Hamilton cycles. It is known by many that the replacement of a vertex with a triangle in a trivalent vertex-transitive graph results in a vertex-transitive graph if and only if the original graph is also arc-transitive. In this paper, we generalize this notion to t -regular graphs Γ and replace each vertex with a complete graph K t on t vertices. We determine necessary and sufficient conditions for T (Γ) to be hamiltonian, show Aut( T (Γ)) ≅ Aut(Γ) , as well as show that if Γ is vertex-transitive, then T (Γ ) is vertex-transitive if and only if Γ is arc-transitive. Finally, in the case where t is prime we determine necessary and sufficient conditions for T (Γ) to be isomorphic to a Cayley graph as well as an additional necessary and sufficient condition for T (Γ) to be vertex-transitive.

$$Z_{3}$$ Z 3 -Connectivity of Wreath Product of Graphs

Let $$G$$G and $$G'$$G? be two simple connected nontrivial graphs. Denote by $$G\Box G^{'}$$G?G? and $$G\rho G^{'}$$G?G? the Cartesian product and the Wreath product of $$G$$G and $$G^{'}$$G?, respectively. In this paper, we will show two results: (1) if $$G^{'}$$G? is a triangularly connected simple graph with $$|V(G^{'})|\ge 3$$|V(G?)|?3 which is neither $$K_{2,t}^{+}$$K2,t+ nor one specific graph, then $$G\Box G^{'}$$G?G? is $$Z_{3}$$Z3-connected; (2) if $$G^{'}$$G? is a vertex-transitive graph, then $$G\rho G^{'}$$G?G? is $$Z_{3}$$Z3-connected.

Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

The main result of this paper is that, if Γ is a connected 4-valent G-arc-transitive graph and v is a vertex of Γ, then either Γ is part of a well-understood infinite family of graphs, or |Gv|≤2436 or 2|Gv|log2⁡(|Gv|/2)≤|VΓ| and that this last bound is tight. As a corollary, we get a similar result for 3-valent vertex-transitive graphs.

Vertex-transitive CIS graphs

A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and only if it is well-covered, co-well-covered, and the product of its clique and stability numbers equals its order. A graph is irreducible if no two distinct vertices have the same neighborhood. We classify irreducible well-covered CIS graphs with clique number at most 3 and vertex-transitive CIS graphs of valency at most 7, which include an infinite family. We also exhibit an infinite family of vertex-transitive CIS graphs which are not Cayley.

Finite Locally-quasiprimitive Graphs

A graph 𝛤 is called G-locally-quasiprimitive if each normal subgroup of Gv acts on 𝛤(v) trivially or transitively for every vertex v. In this paper we analyse the global action and the structural information of such groups G, extending the previous results for locally-primitive graphs and vertex-transitive locally-quasiprimitive graphs.

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.

Polarity graphs revisited

Polarity graphs, also known as Brown graphs, and their minor modifications are the largest currently known graphs of diameter 2 and a given maximum degree d such that d − 1 is a prime power larger than 5 . In view of the recent interest in the degree-diameter problem restricted to vertex-transitive and Cayley graphs we investigate ways of turning the (non-regular) polarity graphs to large vertex-transitive graphs of diameter 2 and given degree. We review certain properties of polarity graphs, giving new and shorter proofs. Then we show that polarity graphs of maximum even degree d cannot be spanning subgraphs of vertex-transitive graphs of degree at most d + 2 . If d − 1 is a power of 2 , there are two large vertex-transitive induced subgraphs of the corresponding polarity graph, one of degree d − 1 and the other of degree d − 2 . We show that the subgraphs of degree d − 1 cannot be extended to vertex-transitive graphs of diameter 2 by adding a relatively small non-edge orbital. On the positive side, we prove that the subgraphs of degree d − 2 can be extended to the largest currently known Cayley graphs of given degree and diameter 2 found by Šiagiová and the second author [ J. Combin. Theory Ser. B 102 (2012), 470–473].