Spectral Characterization of the Hamming Graphs

Theoretical Computer Science, 2008

This paper presents strategies for improving the known upper and lower bounds for the bandwidth of Hamming graphs (K n ) d and [0, 1] d . In particular, it is shown that the the bandwidth of K 6 × K 6 × K 6 is exactly 101. The same numbering strategy lowers the upper bound on the bandwidth of the continuous Hamming graph, [0, 1] 3 , from .5 to .4497. A lower bound of .4439 on bw([0, 1] 3 ) follows from known isoperimetric inequalities and a related dynamic program is conjectured to raise that lower bound to 4/9 = .4444....

arXiv (Cornell University), 2024

The Q-polynomial property is an algebraic property of distance-regular graphs, that was introduced by Delsarte in his study of coding theory. Many distance-regular graphs admit the Q-polynomial property. Only recently the Q-polynomial property has been generalized to graphs that are not necessarily distance-regular. In [21] it was shown that graphs arising from the Hasse diagrams of the so-called attenuated space posets are Q-polynomial. These posets could be viewed as q-analogs of the Hamming posets, which were not studied in [21]. The main goal of this paper is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are Q-polynomial.

The D-eigenvalues {µ 1 , µ 2 , . . . , µ p } of a graph G are the eigenvalues of its distance matrix D and form the distance spectrum or D-spectrum of G denoted by spec D (G). In this paper we obtain the D-spectrum of the cartesian product of two distance regular graphs. The D-spectrum of the lexicographic product G[H] of two graphs G and H when H is regular is also obtained. The D-eigenvalues of the Hamming graphs Ham(d, n) of diameter d and order n d and those of the C 4 nanotori, T k,m,C4 are determined.

European Journal of Combinatorics, 1996

This paper contains a new algorithm that recognizes whether a given graph G is a Hamming graph , i . e . a Cartesian product of complete graphs , in O ( m ) time and O ( n 2 ) space . Here m and n denote the numbers of edges and vertices of G , respectively . Previously this was only possible in O ( m log n ) time .

A coloring of the vertices of a graph G is a distance k coloring of G if and only if any two vertices lying on a path of length less than or equal to k are given dierent colors. Hamming graphs are Cartesian (or box) products of complete graphs. In this paper, we will consider the interaction between coding theory and distance k colorings of Hamming graphs.

European Journal of Combinatorics, 2009

It is proven that given G a subdivision of a clique K n (n ≥ 1), G is isometrically embeddable in a Hamming graph if and only if G is a partial cube or G = K n . The characterization for subdivided wheels is also obtained.

Discrete Mathematics, 1996

The k-spectrum st(G ) of a graph G is the set of all positive integers that occur as the size of an induced k-vertex subgraph of G. In this paper we determine the minimum order and size of a graph G with s k(G) = {0, 1 ..... (~)} and consider the more general question of describing those sets S ~_ [0, 1 ..... (~)} such that S = Sk(G)for some graph G.

2019

Graph embeddings deal with injective maps from a given simple, undirected graph G=(V,E) into a metric space, such as R^n with the Euclidean metric. This concept is widely studied in computer science, see <cit.>, but also offers attractive research in pure graph theory <cit.>. In this note we show that any graph can be embedded into a particularly simple metric space: {0,1}^n with the Hamming distance, for large enough n.

Discrete Applied Mathematics, 1999

We consider the graphs H n a de ned as the Cartesian products of n complete graphs with a vertices each. Let an edge cut partition the vertex set of a graph into k subsets A 1 ; : : : ; A k with jjA i j ? jA j jj 1. We consider the problem of determining the minimal size of such a cut for the graphs de ned above and present bounds and asymptotic results for some speci c values of k.

2011

A cograph is a P 4-free graph. We first give a short proof of the fact that 0 (−1) belongs to the spectrum of a connected cograph (with at least two vertices) if and only if it contains duplicate (resp. coduplicate) vertices. As a consequence, we next prove that the polynomial reconstruction of graphs whose vertex-deleted subgraphs have the second largest eigenvalue not exceeding √ 5−1 2 is unique. 1 MSC: 05C50.

The Kite graph, denoted by $Kite_{p,q}$ is obtained by appending a complete graph $K_p$ to a pendant vertex of a path $P_q$. In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t the adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and let $w(G)$ be the clique number of $G$. Then, it is shown that $w(G) ≥ p − 2q + 1$. Also, we prove that $Kite_{p,2}$ graphs are determined by their adjacency spectrum.

2018

Hamming distance of a two bit strings u and v of length n is defined to be the number of positions of u and v with different digit. If G is a simple graph on n vertices and m edges and B is an edge–vertex incidence matrix of G, then every edge e of G can be labeled using a binary digit string of length n from the row of B which corresponds to the edge e. We discuss Hamming distance of two different edges of the graph G. Then, we present formulae for the sum of all Hamming distances between two different edges of G, particularly when G is a path, a cycle, and a wheel, and some composite graphs.

Electronic Notes in Discrete Mathematics, 2006

A generalized Hamming graph is a graph obtained by a cartesian product of different Hamming graphs. We provide in this paper, under some conditions, a characterization of these graphs using the automorphism group. Then, to study the vertex transitivity of some quasi-amply regular graphs, we give in first the properties of the automorphism group of a prime amply-regular graphs; then we exploit, particulary the main properties of the cartesian product, the prime factor decomposition and the notion of relatively prime graphs. Some graphs of abelian transitive Automorphism group are finally considered.

In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as random-ized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let Kw denote a complete graph on w vertices. In the paper, we show that multicone graphs Kw LHS and Kw LGQ(3, 9) are determined by both their adjacency spectra and their Lapla-cian spectra, where LHS and LGQ(3, 9) denote the Local Higman–Sims graph and the Local GQ(3, 9) graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra.

Universitext, 2012

Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. And the theory of association schemes and coherent configurations studies the algebra generated by associated matrices. Spectral graph theory is a useful subject. The founders of Google computed the Perron-Frobenius eigenvector of the web graph and became billionaires. The second largest eigenvalue of a graph gives information about expansion and randomness properties. The smallest eigenvalue gives information about independence number and chromatic number. Interlacing gives information about substructures. The fact that eigenvalue multiplicities must be integral provides strong restrictions. And the spectrum provides a useful invariant. This book gives the standard elementary material on spectra in Chapter 1. Important applications of graph spectra involve the largest or second largest or smallest eigenvalue, or interlacing, topics that are discussed in Chapters 3-4. Afterwards, special topics such as trees, groups and graphs, Euclidean representations, and strongly regular graphs are discussed. Strongly related to strongly regular graphs are regular two-graphs, and Chapter 10 mainly discusses Seidel's work on sets of equiangular lines. Strongly regular graphs form the first nontrivial case of (symmetric) association schemes, and Chapter 11 gives a very brief introduction to this topic, and Delsarte's Linear Programming Bound. Chapter 12 very briefly mentions the main facts on distance-regular graphs, including some major developments that occurred since the monograph [51] was written (proof of the Bannai-Ito conjecture, construction by Van Dam & Koolen of the twisted Grassmann graphs, determination of the connectivity of distance-regular graphs). Instead of working over R, one can work over F p or Z and obtain more detailed information. Chapter 13 considers pranks and Smith Normal Forms. Finally, Chapters 14 and 15 return to the real spectrum and consider in what cases a graph is determined by its spectrum, and when it has only few eigenvalues. v vi Preface Royle [169]. For association schemes and distance-regular graphs, see Bannai & Ito [19] and Brouwer, Cohen & Neumaier [51].