We study functions defined on the vertices of the Hamming graphs H(n,q). The adjacency matrix of H(n,q) has n+1 distinct eigenvalues n(q−1)−q⋅i with corresponding eigenspaces Ui(n,q) for 0≤i≤n. In ...this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum Ui(n,q)⊕Ui+1(n,q)⊕⋯⊕Uj(n,q) for 0≤i≤j≤n. For the case i+j≤n and q≥3 we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case i+j>n and q≥4 we also find the minimum cardinality of the support of functions, and obtain a characterization of functions with the minimum cardinality of the support for i=j, i>n2 and q≥5. In particular, we characterize eigenfunctions from the eigenspace Ui(n,q) with the minimum cardinality of the support for cases i≤n2, q≥3 and i>n2, q≥5.
Let r≥0 be a fixed integer. If F is an edge set of a connected graph G satisfying the minimum degree of G−F being at least r, then F is a conditional faulty edge set of order r. The graph G is called ...F-strongly Menger-edge-connected if each pair of vertices u and v are connected by min{degG−F(u),degG−F(v)} edge-disjoint paths in G−F, where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. A graph G is m-strongly Menger-edge-connected of order r if G is F-strongly Menger-edge-connected of order r for every F⊂E(G) with |F|≤m and F is a conditional edge fault set of order r, and the maximum value of m is written as smλr(G). Hamming graph KLn has been widely concerned by researchers due to its excellent properties such as good connectivity, scalability, symmetry and iterativity. This paper considers various sufficient conditions of KLn to be F-strongly Menger-edge-connected of order r and determine the exact value of smλ(L−1)r(KLn) by studying the edge-disjoint paths in KLn with edge faults, smλ(L−1)r(KLn)=(L−1)Lr(n−r)−(L−1)n, where 0≤r≤n−2 and n≥3.
Let G be a graph. The 2-token graphF2(G) of G is with vertex set all the 2-subsets of V(G) such that two 2-subsets are adjacent if their symmetric difference is exactly an edge of G. In this paper, ...the full automorphism group of the 2-token graph of the Hamming graph is determined.
We introduce the notion of Levenshtein graphs, an analog to Hamming graphs but using the edit distance instead of the Hamming distance; in particular, vertices in Levenshtein graphs may be strings ...(i.e., words or sequences of characters in a reference alphabet) of possibly different lengths. We study various properties of these graphs, including a necessary and sufficient condition for their shortest path distance to be identical to the edit distance, and characterize their automorphism group and determining number. We also bound the metric dimension (i.e. minimum resolving set size) of Levenshtein graphs. Regarding the latter, recall that a run is a string composed of identical characters. We construct a resolving set of two-run strings and an algorithm that computes the edit distance between a string of length k and any single-run or two-run string in O(k) operations.
A k-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than k in the graph. The k-independence number of a graph, denoted αk, is ...the size of a largest k-independent set in the graph. Recent results have made use of polynomials that depend on the spectrum of the graph to bound the k-independence number. They are optimized for the cases k=1,2. There are polynomials that give good (and sometimes) optimal results for general k, including case k=3. In this paper, we provide the best possible bound that can be obtained by choosing a polynomial for case k=3 and apply this bound to well-known families of graphs including the Hamming graph.
•In this paper, we characterize Hamming and Johnson graphs by their spectral gap.•We confirm Babai's conjecture on the motion of distance-regular graphs of bounded diameter.•Our characterization of ...Hamming graphs involves only inequality constraints.
One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with μ≥2 and second largest eigenvalue θ1=b1−1. In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ1≥(1−ε)b1 for the class of geometric distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism group of distance-regular graphs of bounded diameter. This conjecture asserts that if X is a primitive distance-regular graph with n vertices, and X is not a Hamming graph or a Johnson graph, then the automorphism group Aut(X) has minimal degree ≥cn for some constant c>0. It follows that Aut(X) satisfies strong structural constraints.
We prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More ...generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes.
For every graph
G $G$, let
α
(
G
) $\alpha (G)$ denote its independence number. What is the minimum of the maximum degree of an induced subgraph of
G $G$ with
α
(
G
)
+
1 $\alpha (G)+1$ vertices? We ...study this question for the
n $n$‐dimensional Hamming graph over an alphabet of size
k $k$. In this paper, we give a construction to prove that the answer is 1 for all
n $n$ and
k $k$ with
k
≥
3 $k\ge 3$. This is an improvement over an earlier work showing that the answer is at most
⌈
n
⌉ $\lceil \sqrt{n}\rceil $.
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