Infinite Families Of Such Graphs Research Articles

Semisymmetric Graphs of Order 2 p3 with Valency p2

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph [Formula: see text] of order [Formula: see text], [Formula: see text] a prime, it is well known that [Formula: see text] is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group [Formula: see text] acting unfaithfully on at least one bipart of [Formula: see text], which shows that there is only one infinite family of such graphs with valency [Formula: see text]. The graphs of this kind have been determined when [Formula: see text] acts faithfully and primitively on at least one bipart of [Formula: see text], and thus there is only one remaining case for classifying such graphs of valency [Formula: see text], [Formula: see text] acting faithfully and imprimitively on both biparts of [Formula: see text], which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order [Formula: see text] with valency [Formula: see text].

On the connectivity and diameter of geodetic graphs

A graph G is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We may assume, of course, that G is 2-connected, and here we consider only graphs with no vertices of degree 1 or 2. We prove that all such graphs are, in fact 3-connected. We also construct an infinite family of such graphs of the largest known diameter, namely 5.

A characterization of 4-χS-vertex-critical graphs for packing sequences with s1=1 and s2≥3

If S=(s1,s2,…) is a non-decreasing sequence of positive integers, then the S-packing k-coloring of a graph G is a mapping c:V(G)→[k] such that if c(u)=c(v)=i for u≠v∈V(G), then dG(u,v)>si. The S-packing chromatic number of G is the smallest integer k such that G admits an S-packing k-coloring. A graph G is χS-vertex-critical if χS(G−u)<χS(G) for each u∈V(G). If G is χS-vertex-critical and χS(G)=k, then G is k−χS-vertex-critical. In this paper, 4−χS-vertex-critical graphs are characterized for sequences S=(1,s2,s3,…) with s2≥3. There are 28 sporadic examples and two infinite families of such graphs.

Intersection density of cubic symmetric graphs

Two elements g, h of a permutation group G acting on a set V are said to be intersecting if g(v) = h(v) for some v in V. More generally, a subset mathcal {F} of G is an intersecting set if every pair of elements of mathcal {F} is intersecting. The intersection density rho (G) of a transitive permutation group G is the maximum value of the quotient |mathcal {F}|/|G_v| where mathcal {F} runs over all intersecting sets in G and G_v is the stabilizer of v in V. A vertex-transitive graph X is intersection density stable if any two transitive subgroups of {textrm{Aut}},(X) have the same intersection density. This paper studies the above concepts in the context of cubic symmetric graphs. While a 1-regular cubic symmetric graph is necessarily intersection density stable, the situation for 2-arc-regular cubic symmetric graphs is more complex. A necessary condition for a 2-arc-regular cubic symmetric graph admitting a 1-arc-regular subgroup of automorphisms to be intersection density stable is given, and an infinite family of such graphs is constructed.

Fractional revival on non-cospectral vertices

Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices a and b in a graph, there cannot be perfect state transfer from a to any other vertex. No examples of unweighted graphs with fractional revival between non-cospectral vertices were known; here we give an infinite family of such graphs. No examples of unweighted graphs where the pairs involved in fractional revival overlapped were known; we give examples of such graphs as well.

Planar hypohamiltonian oriented graphs

AbstractIn 1978 Thomassen asked whether planar hypohamiltonian oriented graphs exist. Infinite families of such graphs have since been described but for infinitely many it remained an open question whether planar hypohamiltonian oriented graphs of order exist. In this paper we develop new methods for constructing hypohamiltonian digraphs, which, combined with efficient graph generation algorithms, enable us to fully characterise the orders for which planar hypohamiltonian oriented graphs exist. Our novel methods also led us to discover the planar hypohamiltonian oriented graph of smallest order and size, as well as infinitely many hypohamiltonian orientations of maximal planar graphs. Furthermore, we answer a question related to a problem of Schiermeyer on vertex degrees in hypohamiltonian oriented graphs, and characterise all the orders for which planar hypotraceable oriented graphs exist.

Bipartite biregular Moore graphs

A bipartite graph G=(V,E) with V=V1∪V2 is biregular if all the vertices of a stable set Vi have the same degree ri for i=1,2. In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol, and Fàbrega.Besides, we propose some constructions of bipartite biregular graphs with diameter d and large number of vertices N(r1,r2;d), together with their spectra. In some cases of diameters d=3, 4, and 5, the new graphs attaining the Moore bound are unique up to isomorphism.

Edge‐maximal graphs on orientable and some nonorientable surfaces

AbstractWe study edge‐maximal, noncomplete graphs on surfaces that do not triangulate the surface. We prove that there is no such graph on the projective plane is the unique such graph on the Klein bottle and is the unique such graph on the torus . In contrast to this for each we construct an infinite family of such graphs on the orientable surface of genus , that are edges short of a triangulation.

Solutions to four open problems on quorum colorings of graphs

A partition π = {V1, V2,…,Vk} of the vertex set V of a graph G into k color classes Vi, with 1 ≤ i ≤ k is called a quorum coloring if for every vertex v ∈ V, at least half of the vertices in the closed neighborhood N [v] of v have the same color as v. The maximum cardinality of a quorum coloring of G is called the quorum coloring number of G and is denoted ψq (G). In this paper, we give answers to four open problems stated in 2013 by Hedetniemi, Hedetniemi, Laskar and Mulder. In particular, we show that there is no good characterization of the graphs G with ψq (G) nor for those with ψq (G) &gt; 1 unless 𝒫 ≠ 𝒩𝒫 ∩ co – 𝒩𝒫. We also construct several new infinite families of such graphs, one of which the diameter diam (G) of G is not bounded.

Semisymmetric prime-valent graphs of order $$2p^3$$

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime and $$\Gamma $$ a semisymmetric prime-valent graph of order $$2p^3$$ . Then, $$\Gamma $$ is bipartite. Denote by $$\mathrm{Aut}(\Gamma )$$ the full automorphism group of $$\Gamma $$ . In Du and Wang (J Algebraic Combin 41:275–302, 2015), Wang and Du (Eur J Combin 36:393–405, 2014) and Wang et al. (Ars Math Contemp 7:40–53, 2014), the first author and Du proved that there is no prime-valent graph when $$\mathrm{Aut}(\Gamma )$$ acts unfaithfully on at least one bipart of $$\Gamma $$ . The first author in Wang (Ars Combin 133:3–15, 2017) gave a complete classification when $$\mathrm{Aut}(\Gamma )$$ acts faithfully and primitively on at least one bipart of $$\Gamma $$ , and as a result there is only one such graph, that is, cubic Gray graph. Due to these efforts, there is only one remaining case for classifying such graphs: $$\mathrm{Aut}(\Gamma )$$ acts faithfully and imprimitively on both biparts of $$\Gamma $$ , which is dealt with in this paper. We prove that there are two infinite families of such graphs. Thus, combining these results, we can get the complete classification of semisymmetric prime-valent graphs of order $$2p^3$$ .

Highly-connected planar cubic graphs with few or many Hamilton cycles

In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen (2012) about extremal graphs in these families. In particular, we find families of cyclically 5-edge-connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs P(2n,2). The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes—more precisely, the family consists of the zigzag nanotubes of (fixed) width 5and increasing length. In order to count the Hamilton cycles in the nanotubes, we develop methods inspired by the transfer matrices of statistical physics. We outline how these methods can be adapted to count the Hamilton cycles in nanotubes of greater (but still fixed) width, with the caveat that the resulting expressions involve matrix powers. We also consider cyclically 4-edge-connected planar cubic graphs with few Hamilton cycles, and exhibit an infinite family of such graphs each with exactly 4 Hamilton cycles. Finally we consider the “other extreme” for these two classes of graphs, thus investigating cyclically 4-edge-connected planar cubic graphs with many Hamilton cycles and the cyclically 5-edge-connected planar cubic graphs with few Hamilton cycles. In each of these cases, we present partial results, examples and conjectures regarding the graphs with few or many Hamilton cycles.

Hexavalent half-arc-transitive graphs of order [formula omitted

A graph is called half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. In this paper, we classify hexavalent half-arc-transitive graphs of order 9p for each prime p. As a result, there are four infinite families of such graphs: three defined on Zp⋊Z27 with 27|(p−1); one defined on Z3p⋊Z9 with 9|(p−1).

Three-weight codes, triple sum sets, and strongly walk regular graphs

We construct strongly walk-regular graphs as coset graphs of the duals of three-weight codes over $$\mathbb {F}_q.$$ The columns of the check matrix of the code form a triple sum set, a natural generalization of partial difference sets. Many infinite families of such graphs are constructed from cyclic codes, Boolean functions, and trace codes over fields and rings. Classification in short code lengths is made for $$q=2,3,4$$ .

On the girth of two-dimensional real algebraically defined graphs

For a polynomial f∈R[X,Y], we define a bipartite graph ΓR(f) where each partite set is a copy of R2. Furthermore, (a1,a2) in the first partite set is adjacent to [x1,x2] in the second if and only if a2+x2=f(a1,x1). The main result of this paper is that every graph ΓR(f) has girth 4 or 6, and moreover we classify infinite families of such graphs by girth.

Infinite family of 2-connected transmission irregular graphs

Distance between two vertices is the number of edges in the shortest path connecting them in a connected graph G. The transmission of a vertex v is the sum of distances from v to all the other vertices of G. If transmissions of all vertices are mutually distinct, then G is a transmission irregular graph. It is known that almost no graphs are transmission irregular. Infinite families of transmission irregular trees were presented in [4]. The following problem was posed in [4]: do there exist infinite families of 2-connected transmission irregular graphs? In this paper, an infinite family of such graphs is constructed.

Cubic vertex-transitive non-Cayley graphs of order 12p

A graph is said to be vertex-transitive non-Cayley if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order 12p, where p is a prime, is given. As a result, there are 11 sporadic and one infinite family of such graphs, of which the sporadic ones occur when p equals 5, 7 or 17, and the infinite family exists if and only if p ≡ 1 (mod 4), and in this family there is a unique graph for a given order.

Finite Edge-Transitive Oriented Graphs of Valency Four: a Global Approach

We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of `basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

The minimal [formula omitted]-symmetric graphs that are not [formula omitted]-spherical

Given a graph G equipped with faithful and fixed-point-free Γ-action (Γ a finite group) we define an orbit minor H of G to be a minor of G for which the deletion and contraction sets are closed under the Γ-action. The orbit minor H inherits a Γ-symmetry from G, and when the contraction set is acyclic the action inherited by H remains faithful and fixed-point free. When G embeds in the sphere and the Γ-action on G extends to a Γ-action on the entire sphere, we say that G is Γ-spherical. In this paper we determine for every odd value of n≥3 the orbit-minor-minimal graphs G with a faithful and free Zn-action that are not Zn-spherical. There are 11 infinite families of such graphs, each of the 11 having exactly one member for each n. For n=3, another such graph is K3,3. The remaining graphs are, essentially, the Cayley graphs for Zn aside from the cycle of length n. The result for n=1 is exactly Wagner’s result from 1937 that the minor-minimal graphs that are not embeddable in the sphere are K5 and K3,3.

Vertex-transitive generalized Cayley graphs which are not Cayley graphs

The concept of generalized Cayley graphs was introduced by Marušič et al. (1992), where it was asked if there exists a vertex-transitive generalized Cayley graph which is not a Cayley graph. In this paper the question is answered in the affirmative with a construction of two infinite families of such graphs. It is also proven that every generalized Cayley graph admits a semiregular automorphism.

On edge-transitive cubic graphs of square-free order

A regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. In this paper, we give a complete list of connected semisymmetric cubic graph of square-free order, which consists of one single graph of order 210 and four infinite families of such graphs.