Odd Graphs Research Articles

One note on the number of pairwise non-isomorphic connected regular graphs

In this paper, we give an explicit expression for the number of pairwise non-isomorphic connected regular graphs as well as odd graphs on n vertices, where [Formula: see text] is a positive integer.

Neighbour-transitive codes in Kneser graphs

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the first s+1 parts C0,C1,…,Cs of the distance partition{C=C0,C1,…,Cρ}, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph K(n,k) is defined. Our first main result says that if C is a 2-neighbour-transitive code in K(n,k) such that C has minimum distance at least 5, then n=2k+1 (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph K(n,k). First, if Aut(C) acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that Aut(C) acts transitively on Ω, first proving that if C has minimum distance at least 3 then either K(n,k) is an odd graph or Aut(C) has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and Aut(C) acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.

On the spectra of token graphs of cycles and other graphs

The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs Or for all r, and the multipartite complete graphs Kn1,n2,…,nr for all n1,n2,…,nr In the case of cycles, we present a new method that allows us to compute the whole spectrum of F2(Cn). This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of F2(Cn).

Odd edge‐colorings of subdivisions of odd graphs

AbstractAn odd graph is a finite graph all of whose vertices have odd degrees. A graph is decomposable into odd subgraphs if its edge set can be partitioned into subsets each of which induces an odd subgraph of . The minimum value of for which such a decomposition of exists is the odd chromatic index, , introduced by Pyber. For every , the graph is said to be odd ‐edge‐colorable. Apart from two particular exceptions, which are, respectively, odd 5‐ and odd 6‐edge‐colorable, the rest of connected loopless graphs are odd 4‐edge‐colorable, and moreover one of the color classes can be reduced to size . In addition, it has been conjectured that an odd 4‐edge‐coloring with a color class of size at most 1 is always achievable. Atanasov et al. characterized the class of loopless subcubic graphs in terms of the value . In this paper, we extend their result to a characterization of all loopless subdivisions of odd graphs in terms of the value of the odd chromatic index. This larger class is of a particular interest as it collects all “least instances” of nonodd graphs. As a prelude to our main result, we show that every connected graph requiring the maximum number of four colors, becomes odd 3‐edge‐colorable after removing a certain edge. Thus, we provide support for the mentioned conjecture by proving it for all subdivisions of odd graphs. The paper concludes with few problems for possible further work.

Characterization of 2-arc-transitive partial cubes

Let Γ be a graph with its automorphism group G and s(⩾0) an integer. We call a vertex sequence (v0,v1,…,vs) of length s+1 of Γ an s-arc if two consecutive vertices are adjacent and vi≠vi+2 for 0⩽i⩽s−2. Γ is called s-arc-transitive if G acts on its s-arc set transitively. If Γ is s-arc-transitive, but not (s+1)-arc-transitive, we call it an s-transitive graph. Γ is a partial cube if Γ can be embedded into a hypercube isometrically. In this paper, we characterized non-trivial regular 2-arc-transitive partial cubes as three classes of graphs: hypercubes, doubled Odd graphs and even cycles. As a corollary, we prove that there exist no regular s-transitive partial cubes for s⩾4 and characterized regular s-transitive partial cubes for s=2 or 3.

On Total Coloring of Some Classes of Regular Graphs

In this paper, we have obtained the total chromatic number of some classes of Cayley graphs, odd graphs and mock threshold graphs.

2-divisibility of Some Odd Hole Free Graphs

2-divisibility of Some Odd Hole Free Graphs

On the Laplacian spectra of token graphs

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that 1≤h≤k≤n2, the Laplacian spectrum of Fh(G) is contained in the Laplacian spectrum of Fk(G). We also show that the doubled odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph Kn and the star Sn=K1,n−1, respectively. Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement G‾. This generalizes to tokens graphs a well-known property stating that the Laplacian eigenvalues of G are closely related to the Laplacian eigenvalues of G‾. Finally, the doubled odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph G and its token graph.

A combinatorial basis for Terwilliger algebra modules of a bipartite distance-regular graph

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and for any integer i, let Γi(x) denote the set of vertices at distance i from x. Let V=ℂX denote the vector space over ℂ consisting of column vectors whose coordinates are indexed by X and whose entries are in ℂ, and for z∈X let ẑ denote the element of V with a 1 in the z coordinate and 0 in all other coordinates. Fix vertices x,u,v where u∈Γ2(x) and v∈Γ2(x)∩Γ2(u), and let T=T(x) denote the Terwilliger algebra with respect to x. Under certain additional combinatorial assumptions, we give a combinatorially-defined spanning set for a T-module of endpoint 2,and we give the action of the adjacency matrix on this spanning set. The vectors in our spanning set are defined as sums and differences of vectors ẑ, where the vertices z are chosen based on the their distances from x,u, and v.We use this T-module to construct combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 for examples including the Doubled Odd graphs, the Double Hoffman–Singleton graph, Tutte’s 12-cage graph, and the Foster graph. We provide a list of several other graphs satisfying our conditions.

On even-cycle-free subgraphs of the doubled Johnson graphs

The generalized Turán number ex(G,H) is the maximum number of edges in an H-free subgraph of a graph G. It is an important extension of the classical Turán number ex(n,H), which is the maximum number of edges in a graph with n vertices that does not contain H as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs J(n;k,k+1), which are bipartite subgraphs of hypercube graphs. We give an upper bound for ex(J(n;k,k+1),C2r) with any fixed k∈Z+ and any n∈Z+ with n≥2k+1. We also give an upper bound for ex(J(2k+1;k,k+1),C2r) with any k∈Z+, where J(2k+1;k,k+1) is known as doubled Odd graph O˜k+1. This bound induces that the number of edges in any C2r-free subgraph of O˜k+1 is o(e(O˜k+1)) for r≥6, which also implies a Ramsey-type result.

The super-connectivity of odd graphs and of their kronecker double cover

The study of connectivity parameters forms an integral part of the research conducted in establishing the fault tolerance of networks. A number of variations on the classical notion of connectivity have been proposed and studied. In particular, the super-connectivity asks for the minimum number of vertices that need to be deleted from a graph in order to disconnect the graph without creating isolated vertices. In this work, we determine this value for two closely related families of graphs which are considered as good models for networks, namely the odd graphs and their Kronecker double cover. The odd graphs are constructed by taking all possible subsets of size k from the set of integers {1,…,2k + 1} as vertices, and defining two vertices to be adjacent if the corresponding k-subsets are disjoint; these correspond to the Kneser graphs KG(2k + 1, k). The Kronecker double cover of a graph G is formed by taking the Kronecker product of G with the complete graph on two vertices; in the case when G is KG(2k + 1, k), the Kronecker double cover is the bipartite Kneser graph H(2k + 1, k). We show that in both instances, the super-connectivity is equal to 2k.

Sparse Kneser graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words.

Partitioning Transitive Tournaments into Isomorphic Digraphs

In an earlier paper (see Sali and Simonyi Eur. J. Combin. 20, 93–99, 1999) the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.

Integral graphs obtained by dual Seidel switching

Dual Seidel switching is a graph operation introduced by W. Haemers in 1984. This operation can change the graph, however it does not change its bipartite double, and because of this, the operation leaves the squares of the eigenvalues invariant. Thus, if a graph is integral then it is still integral after dual Seidel switching. In this paper two new infinite families of integral graphs are obtained by applying dual Seidel switching to the Star graphs and the Odd graphs. In particular, three new 4-regular integral graphs with their spectra are found.

A new class of polynomials from the spectrum of a graph, and its application to bound the k-independence number

The k-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k. A graph is called k-partially walk-regular if the number of closed walks of a given length l≤k, rooted at a vertex v, only depends on l. In particular, a distance-regular graph is also k-partially walk-regular for any k. In this paper, we introduce a new family of polynomials obtained from the spectrum of a graph, called minor polynomials. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the k-independence number of a k-partially walk-regular graph. With some examples and infinite families of graphs whose bounds are tight, we also show that the odd graph Oℓ with ℓ odd has no 1-perfect code. Moreover, we show that our bound coincides, in general, with the Delsarte's linear programming bound and the Lovász theta number θ, the best ones to our knowledge. In fact, as a byproduct, it is shown that the given bounds also apply for θ and Θ, the Shannon capacity of a graph. Moreover, apart from the possible interest that the minor polynomials can have, our approach has the advantage of yielding closed formulas and, so, allowing asymptotic analysis.

ON LOCALLY-BALANCED 2-PARTITIONS OF SOME CLASSES OF GRAPHS

In this paper we obtain some conditions for the existence of locally-balanced 2-partitions with an open (with a closed) neighborhood of some classes of graphs. In particular, we give necessary conditions for the existence of locallybalanced 2-partitions of even and odd graphs. We also obtain some results on the existence of locally-balanced 2-partitions of rook’s graphs and powers of cycles. In particular, we prove that if \(m,n \geq 2\), then the graph \(K_m \Box K_n\) has a locally-balanced 2-partition with a closed neighborhood if and only if m and n are even. Moreover, all our proofs are constructive and provide polynomial time algorithms for constructing the required 2-partitions.

On the quantum symmetry of distance-transitive graphs

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs H(n,3), the Johnson graphs J(n,2) and the Kneser graphs K(n,2) do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order ≤20 has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the 4×4 rook's graph), where one of them has quantum symmetry and the other one does not.

Further results on even vertex odd mean graphs

An even vertex odd mean labeling of an undirected graph with q edges is a one to one function f from the set of vertices V(G) to the set 0, 2, 4, … , 2q such that the induced function f* from E(G) to the set 1, 3, 5, … , 2q – 1 defined by is a bijection. A graph that admits an even vertex odd mean labeling is called even vertex odd mean graph. In this paper, we prove that D 2 (Pn ), D 2 (Ln ), D 2 ([P 2n , Sm ]), S(Ln ), S(Pn ⊗ K 1 ), S(SLn ), S′(Pn ), S′(k 2,n ), S′(Pn ⊗ K 1 ), S′(Cn ) for n ≡ 0 (mod 4) and S′([P 2n , Sm ]) are even vertex odd mean graphs.

Further Results On Odd Mean Graphs

Further Results On Odd Mean Graphs

Modular graph functions and odd cuspidal functions. Fourier and Poincar\xe9 series

Modular graph functions are SL(2, ℤ)-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus τ. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to Γ∞\\PSL(2, ℤ). The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under τ → − overline{tau} are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space mathfrak{A} w of odd two-loop modular graph functions of weight w. For w ≤ 11 the bound is saturated and we exhibit a basis for mathfrak{A} w.