Graphs Of Valency Research Articles

Two‐sided Group Digraphs and Graphs

AbstractWe study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.

Finite two-distance-transitive graphs of valency 6

A non-complete graph Gamma is said to be (G,2)-distance-transitive if, for i = 1,2 and for any two vertex pairs (u_1,v_1) and (u_2,v_2) with d_Gamma(u_1,v_1) = d_Gamma(u_2,v_2) = i, there exists g in G such that (u_1,v_1)^g=(u_2,v_2). This paper classifies the family of (G,2)-distance-transitive graphs of valency 6 which are not (G,2)-arc-transitive.

Novel Conditions on the Non-normal Cayley Graphs of Valency Six

A Cayley graph X = Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group Aut(X). In this paper, two novel conditions are outlined to identify the non-normal Cayley graphs of valency six. As an application, some non-normal Cayley graphs of valency six on A6 and A5 are obtained.

Nowhere-zero 3-flows in graphs admitting solvable arc-transitive groups of automorphisms

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a nowhere-zero 3-flow.

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.

Finite 2-geodesic-transitive graphs of valency twice a prime

In this paper, we classify the family of connected 2-geodesic-transitive graphs of valency 2p where p is an odd prime.

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.

Elusive groups of automorphisms of digraphs of small valency

A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.

On geodesic transitive graphs

The main purpose of this paper is to investigate relationships between three graph symmetry properties: s-arc transitivity, s-geodesic transitivity, and s-distance transitivity. A well-known result of Weiss tells us that if a graph of valency at least 3 is s-arc transitive then s≤7. We show that for each value of s≤3, there are infinitely many s-arc transitive graphs that are t-geodesic transitive for arbitrarily large values of t. For 4≤s≤7, the geodesic transitive graphs that are s-arc transitive can be explicitly described, and all but two of these graphs are related to classical generalized polygons. Finally, we show that the Paley graphs and the Peisert graphs, which are known to be distance transitive, are almost never 2-geodesic transitive, with just three small exceptions.

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 3-GEODESIC TRANSITIVE BUT NOT 3-ARC TRANSITIVE GRAPHS

AbstractIn this paper, we first prove that for $g\in \{3,4\}$, there are infinitely many 3-geodesic transitive but not 3-arc transitive graphs of girth $g$ with arbitrarily large diameter and valency. Then we classify the family of 3-geodesic transitive but not 3-arc transitive graphs of valency 3 and those of valency 4 and girth 4.

On cyclic regular covers of complete graphs of small order

The paper presents classifications of edge-transitive cyclic regular covers of the complete graphs K5 and K6, and arc-transitive cyclic regular covers of the complete graph K7. Two new infinite families of transitive graphs of valency 4 and 6 are found. As an application, tetravalent edge-transitive graphs of order 5p2 with p a prime are classified.

The Extendability of Matchings in Strongly Regular Graphs

A graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not $3$-extendable. We also show that strongly regular graphs of valency $k\geq 3$ with $\lambda \geq 1$ are $\lfloor k/3\rfloor$-extendable (when $\mu \leq k/2$) and $\lceil \frac{k+1}{4}\rceil$-extendable (when $\mu>k/2$), where $\lambda$ is the number of common neighbors of any two adjacent vertices and $\mu$ is the number of common neighbors of any two non-adjacent vertices. Our results are close to being best possible as there are strongly regular graphs of valency $k$ that are not $\lceil k/2\rceil $-extendable. We show that the extendability of many strongly regular graphs of valency $k$ is at least $\lceil k/2 \rceil -1$ and we conjecture that this is true for all primitive strongly regular graphs. We obtain similar results for strongly regular graphs of odd order.

Arc-transitive graphs of valency 8 have a semiregular automorphism

One version of the polycirculant conjecture states that every vertex-transitive graph has a non-identity semiregular automorphism that is, a non-identity automorphism whose cycles all have the same length. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open valency.

Semiregular automorphisms of edge-transitive graphs

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism: a non-trivial automorphism whose cycles all have the same length. In this paper, we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

Frobenius circulant graphs of valency six, Eisenstein–Jacobi networks, and hexagonal meshes

A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as G=K⋊H with K a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K with connection set S an H-orbit on K generating K, where H is of even order or S consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel K is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its forwarding indices, Wiener indices and minimum gossip time. We also prove that its broadcasting time is equal to its diameter plus two or three. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein–Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein–Jacobi integers. We also prove that larger Eisenstein–Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for 6-valent first-kind Frobenius circulants. As a corollary any Eisenstein–Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein–Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. A distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.

On symmetric and locally finite actions of groups on the quintic tree

In this paper, we determine all faithful, symmetric and locally finite actions of a group on the tree of valency five. As a corollary we complete the classification of the isomorphism types of vertex and edge stabilisers in a group acting symmetrically on a graph of valency five. This builds on work of Weiss and recent work of Feng, Zhou and Feng, Guo. Our approach is to classify the isomorphism types of finite, faithful amalgams of degree (5, 2).

Isomorphic tetravalent cyclic Haar graphs

Let S be a subset of the cyclic group Z n . The cyclic Haar graph H (Z n , S ) is the bipartite graph with color classes Z n + and Z n − , and edges { x + , y − }, where x , y ∈ Z n and y − x ∈ S . In this paper we give sufficient and necessary conditions for the isomorphism of two connected cyclic Haar graphs of valency 4 .

Finite 1-Regular Cayley Graphs of Valency 5

Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency .

Locally arc-transitive graphs of valence {3,4} with trivial edge kernel

In this paper, we consider connected locally G-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighbourhood Γ(u)?Γ(v) is trivial. We find 19 finitely presented groups with the property that any such group G is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence {3,4} on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup G with $G_{uv}^{[1]} = 1$ .