Geometric aspects of 2-walk-regular graphs

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show

On independent star sets in finite graphs

Let G be a finite graph with μ as an eigenvalue of multiplicity k. A star set for μ is a set X of k vertices in G such that μ is not an eigenvalue of G-X. We investigate independent star sets of largest possible size in a variety of situations. We note connections with symmetric designs, codes, strongly regular graphs, and graphs with least eigenvalue -2

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples

Geometric aspects of 2-walk-regular graphs

A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte’s clique bound to 1-walk-regular graphs, Godsil’s multiplicity bound and Terwilliger’s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show

Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups

In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs $P(n)$ and to a recent known family of mixed graphs $\Gamma(d,n,r)$ (having edges with and without direction). As a byproduct, the existence of perfect codes in $P(n)$ allows us to give a lower bound for the multiplicity of its eigenvalue $-1$

Support of Closed Walks and Second Eigenvalue Multiplicity of the Normalized Adjacency Matrix

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ for simple $d$-regular graphs when $d\ge \log^{1/4}n$. In fact, the same bounds hold for the number of eigenvalues in any interval of width $\lambda_2/\log_\Delta^{1-o(1)}n$ containing the second eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the typical support of a closed random walk of length $2k$ in any connected graph, which in turn relies on new lower bounds for the entries of the Perron eigenvector of submatrices of the normalized adjacency matrix.Comment: A previous version of this paper proved the main result for d-regular graphs. The current version proves a more general result for the normalized adjacency matrix of bounded degree graphs. 24pp, 3 figure

Determination of (0,2)-regular sets in graphs

An eigenvalue of a graph is main iff its associated eigenspace is not orthogonal to the all-one vector j. The main characteristic polynomial of a graph G with p main distinct eigenvalues is _ (λ)=λ^−_0 λ^(−1)−_1 λ^(−2)-…-_(−2) λ−_(−1) and it has integer coefficients. If G has n vertices, the nxk walk matrix of G is _=(j,_j,_^"2" "j",…,_^(−) j) and W, the walk matrix of G, is _ for which rank(_)=k. The number k coincides with the number of distinct main eigenvalues of G. In [2] it was proved that the coefficients of the main characteristic polynomial of G are the solutions of =_^j. A (,)- regular set [3] is a subset of the vertices of a graph inducing a -regular subgraph such that every vertex not in the subset has neighbors in it. In [1], a strategy for the determination of (0,1)-regular sets is described and we generalize it in order to solve the problem of the determination of (0,2)-regular sets in arbitrary graphs. An algorithm for deciding whether or not a given graph has a (0,2)-regular set is described. Its complexity depends on the multiplicity of −2 as an eigenvalue of the adjacency matrix of the graph. When such multiplicity is low, the generalization of the results in [1] assure that the algorithm is polynomial. An example of application of the algorithm to a graph for which this multiplicity is low is also presented

Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3

AbstractIn this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ=−1 if and only if Γ is antipodal. Then we assume that the graph Γ is primitive. We show that it is formally self-dual (and hence Q-polynomial and 1-homogeneous), all its eigenvalues are integral, and the eigenvalue with multiplicity equal to the valency is either second largest or the smallest.Let x,y∈VΓ be two adjacent vertices, and z∈Γ2(x)∩Γ2(y). Then the intersection number τ2≔|Γ(z)∩Γ3(x)∩Γ3(y)| is independent of the choice of vertices x, y and z. In the case of the coset graph of the doubly truncated binary Golay code, we have b2=τ2. We classify all the graphs with b2=τ2 and establish that the just mentioned graph is the only example. In particular, we rule out an infinite family of otherwise feasible intersection arrays

Regular graphs with a complete bipartite graph as a star complement

Let $G$ be a graph of order $n$ and $\mu$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $\mu$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $\mu$, and the vertex subset $X=V(G-H)$ is called a star set for $\mu$ in $G$. The study of star complements and star sets provides a strong link between graph structure and linear algebra. In this paper, we study the regular graphs with $K_{t,s}\ (s\geq t\geq 2)$as a star complement for an eigenvalue$\mu$, especially, characterize the case of$t=3$completely, obtain some properties when$t=s$, and propose some problems for further study

The Support of Open Versus Closed Random Walks

A closed random walk of length ? on an undirected and connected graph G = (V,E) is a random walk that returns to the start vertex at step ?, and its properties have been recently related to problems in different mathematical fields, e.g., geometry and combinatorics (Jiang et al., Annals of Mathematics \u2721) and spectral graph theory (McKenzie et al., STOC \u2721). For instance, in the context of analyzing the eigenvalue multiplicity of graph matrices, McKenzie et al. show that, with high probability, the support of a closed random walk of length ? ? 1 is ?(?^{1/5}) on any bounded-degree graph, and leaves as an open problem whether a stronger bound of ?(?^{1/2}) holds for any regular graph. First, we show that the support of a closed random walk of length ? is at least ?(?^{1/2} / ?{log n}) for any regular or bounded-degree graph on n vertices. Secondly, we prove for every ? ? 1 the existence of a family of bounded-degree graphs, together with a start vertex such that the support is bounded by O(?^{1/2}/?{log n}). Besides addressing the open problem of McKenzie et al., these two results also establish a subtle separation between closed random walks and open random walks, for which the support on any regular (or bounded-degree) graph is well-known to be ?(?^{1/2}) for all ? ? 1. For irregular graphs, we prove that even if the start vertex is chosen uniformly, the support of a closed random walk may still be O(log ?). This rules out a general polynomial lower bound in ? for all graphs. Finally, we apply our results on random walks to obtain new bounds on the multiplicity of the second largest eigenvalue of the adjacency matrices of graphs