Walk-regular Graphs Research Articles

A walk-regular graph, cospectral to its complement, need not be strongly regular

We briefly discuss counterexamples to the conjecture of Lepović (2010) [6]] that a walk-regular graph, which is cospectral to its complement, has to be strongly regular.

On strongly walk regular graphs, triple sum sets and their codes

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k> theta _1> theta _2 > theta _3, and that the triple (theta _1, theta _2, theta _3) satisfies a certain homogeneous polynomial equation of degree s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s = 5 and s = 7, there are only the obvious solutions, and we conjecture this to remain true for all (odd) s ge 9.

Three-Weight Codes over Rings and Strongly Walk Regular Graphs

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $${\mathbb {Z}}_{p^m},$$ for p a prime, and more generally over chain rings of depth m, and with a residue field of size q, a prime power. In the case of $$p=m=2,$$ strong necessary conditions ( diophantine equations) on the weight distribution are derived, leading to a partial classification in modest length. Infinite families of examples are built from Kerdock and generalized Teichmüller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear.

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.

The Edge-Connectivity of Strongly 3-Walk-Regular Graphs

E.R. van Dam and G.R. Omidi generalized the concept of strongly regular graphs as follows. If for any two vertices the number of $$\ell $$-walks (walks of length $$\ell $$) from one vertex to the other is the same which depends only on whether the two vertices are the same, adjacent or non-adjacent, then G is called a strongly $$\ell $$-walk-regular graph. The existence of strongly $$\ell $$-walk-regular graphs which are not strongly 3-walk-regular graphs is unknown. In this paper, we prove that the edge-connectivity of a connected strongly 3-walk-regular graph G of degree $$k\ge 3$$ is equal to k. Moreover, if G is not the graph formed by adding a perfect matching between two copies of $$K_{4}$$, then each edge cut set of size k is precisely the set of edges incident with a vertex of G. For a regular graph G in general, we also give a sufficient and tight condition such that G is 1-extendable.

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 $$q=2,3,4$$ .

Subgraph centrality and walk-regularity

Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between subgraph centrality, which is defined using the matrix exponential f(x)=exp⁡(x), and the walk structure of a network. In a walk-regular graph, the number of closed walks of each length must be the same for all nodes, implying uniform f-subgraph centralities for any f (or maximum f-walk entropy). We consider when non-walk-regular graphs can achieve maximum entropy, calling such graphs entropic. For parameterized measures, we are also interested in which values of the parameter witness this uniformity. To date, only one entropic graph has been identified, with only two witnessing parameter values, raising the question of how many such graphs and parameters exist. We resolve these questions by constructing infinite families of entropic graphs, as well as a family of witnessing parameters with a limit point at zero.

Directed strongly walk-regular graphs

We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly ell -walk-regular with ell > 1 if the number of walks of length ell from a vertex to another vertex depends only on whether the first vertex is the same as, adjacent to, or not adjacent to the second vertex. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph varGamma with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of ell for which varGamma can be strongly ell -walk-regular, and we also construct examples of such strongly walk-regular digraphs. We also consider digraphs with non-real eigenvalues. We give such examples and characterize those digraphs varGamma for which there are infinitely many ell for which varGamma is strongly ell -walk-regular.

2-Walk-Regular Dihedrants from Group-Divisible Designs

In this note, we construct bipartite $2$-walk-regular graphs with exactly 6 distinct eigenvalues as the point-block incidence graphs of group divisible designs with the dual property. For many of them, we show that they are 2-arc-transitive dihedrants. We note that some of these graphs are not described in Du et al. (2008), in which they classified the connected 2-arc transitive dihedrants.

New bounds for the max-k-cut and chromatic number of a graph

We consider several semidefinite programming relaxations for the max-k-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-k-cut when k>2 that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in general.We prove that the eigenvalue bound for the max-k-cut is tight for several classes of graphs. We investigate the presented bounds for specific classes of graphs, such as walk-regular graphs, strongly regular graphs, and graphs from the Hamming association scheme.

Strongly walk-regular graphs

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an ℓ>1 such that the number of walks of length ℓ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly ℓ-walk-regular for all ℓ, whereas the graphs from the fourth family are ℓ-walk-regular for every odd ℓ. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly ℓ-walk-regular for even ℓ. We will characterize the case that regular four-eigenvalue graphs are strongly ℓ-walk-regular for every odd ℓ, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly ℓ-walk-regular for at most one ℓ. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly ℓ-walk-regular for only one particular ℓ different from 3.

On perturbations of almost distance-regular graphs

In this paper we show that certain almost distance-regular graphs, the so-called h-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph G with diameter D is called h-punctually walk-regular, for a given h ≤ D , if the number of paths of length ℓ between a pair of vertices u , v at distance h depends only on ℓ . The graph perturbations considered here are deleting a vertex, adding a loop, adding a pendant edge, adding/removing an edge, amalgamating vertices, and adding a bridging vertex. We show that for walk-regular graphs some of these operations are equivalent, in the sense that one perturbation produces cospectral graphs if and only if the others do. Our study is based on the theory of graph perturbations developed by Cvetković, Godsil, McKay, Rowlinson, Schwenk, and others. As a consequence, some new characterizations of distance-regular graphs are obtained.

Some Results on Walk Regular Graphs Which Are Cospectral to Its Complement

We say that a regular graph $G$ of order $n$ and degree $r\ge 1$ (which is not the complete graph) is strongly regular if there exist non-negative integers $\tau$ and $\theta$ such that $|S_i\cap S_j| = \tau$ for any two adjacent vertices $i$ and $j$, and $|S_i\cap S_j| = \theta$ for any two distinct non-adjacent vertices $i$ and $j$, where $S_k$ denotes the neighborhood of the vertex $k$. We say that a graph $G$ of order $n$ is walk regular if and only if its vertex deleted subgraphs $G_i = G\smallsetminus i$ are cospectral for $i = 1,2,\ldots ,n$. We here establish necessary and sufficient conditions under which a walk regular graph $G$ which is cospectral to its complement $\overline G$ is strongly regular.

Some new results on walk regular graphs which are cospectral to its complement

We say that a regular graph G of order n and degree r ≥ 1 (which is not the complete graph) is strongly regular if there exist non-negative integers τ and θ such that | S i ∩ S j | = τ for any two adjacent vertices i and j , and | S i ∩ S j | = θ for any two distinct non-adjacent vertices i and j , where S k denotes the neighborhood of the vertex k . We say that a graph G of order n is walk regular if and only if its vertex deleted subgraphs G i = G ∖︀ i are cospectral for i = 1 , 2 , … , n . Let G be a walk regular graph of order 4 k + 1 and degree 2 k which is cospectral to its complement G ¯ . Let H i be switching equivalent to G i with respect to S i ⊆ V ( G i ) . We here prove that G is strongly regular if and only if Δ ( G i ) = Δ ( H i ) for i = 1 , 2 , … , 4 k + 1 , where Δ ( G ) is the number of triangles of a graph G .

On t-Cliques in k-Walk-Regular Graphs

Abstract A graph is walk-regular if the number of cycles of length l rooted at a given vertex is a constant through all the vertices. For a walk-regular graph G with d + 1 different eigenvalues and spectrally maximum diameter D = d , we study the geometry of its d-cliques, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a regular tetrahedron and we compute its parameters. Moreover, the results are generalized to the case of k-walk-regular graphs, a family which includes both walk-regular and distance-regular graphs, and their t-cliques or vertices at distance t from each other.

Some Families of Orthogonal Polynomials of a Discrete Variable and their Applications to Graphs and Codes

We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the predistance polynomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polynomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.

On $k$-Walk-Regular Graphs

Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.

Spectral and Geometric Properties of k-Walk-Regular Graphs

Abstract Let us consider a connected graph G with diameter D. For a given integer k between 0 and D, we say that G is k-walk-regular if the number of walks of length l between vertices u, v only depends on the distance between u and v, provided that such a distance does not exceed k. Thus, in particular, a 0-walk-regular graph is the same as a walk-regular graph, where the number of cycles of length l rooted at a given vertex is a constant through all the graph. In the other extreme, the distance-regular graphs correspond to the case k = D . In this talk we discuss some algebraic characterizations of k-walk-regularity, in terms of the local spectrum and pre-distance-polynomials of G. Moreover, some results on the relationship between the diameter and the spectrum, as well as some geometric properties, of walk-regular graphs are presented.

Feasibility conditions for the existence of walk-regular graphs

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.