Recent work has used variations of the hypergraph eigenvalues we describe to obtain results about the maximal cliques in a hypergraph [6], cliques in The nullity of a graph G, denoted by (G), is the multiplicity of the eigenvalue zero in its spectrum. On the hull sets and hull number of the cartesian product of graphs My try: Let | V ( G) | = The second largest eigenvalue of a graph (a survey). : Expander graphs and coding theory.

For two disjoint graphs and , the strong product of them is written as , that is, , and two distinct vertices and are contiguous. The eigenvalue based methods have proved to be useful also for some other problems, e.g. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. The hypercube has been considered in parallel computers, (Ncube, iPSC/860, TMC CM-2, etc.) We show several results about edge-tenacious graphs as well asfind numerous classes of edge-tenacious graphs.The Cartesian Products Now if the vectors are of unit length, ie if they have been standardized, then the dot product of the vectors is equal to cos , and we can reverse calculate from the dot product. In this section, we give a new general method for constructing integral graphs using the Kronecker product and commuting sets of matrices with integral eigenvalues. The Cartesian product G x H of graphs G and H Let C be the adjacency matrix for the Cartesian product H1 H2. 2 Eigenvalues of graphs 2.1 Matrices associated with graphs We introduce the adjacency matrix, the Laplacian and the transition matrix of the random walk, and their eigenvalues. It is also well-known [9, Lemma 13.1.3] that if Ghas no multiple 119 Product dimension. DOI: 10.1016/J.LAA.2010.10.026 Corpus ID: 119584545; On products and line graphs of signed graphs, their eigenvalues and energy @article{Germina2010OnPA, title={On products and line graphs of signed graphs, their eigenvalues and energy}, author={K. Augustine Germina and K ShahulHameed and Thomas Zaslavsky}, journal={Linear Algebra mk , 2010 Mathematics Subject Classication. f198 GHORBANI, SEYED-HADI AND NOWROOZI-LARKI By a circulant matrix, we mean a square nn matrix whose rows are a cyclic permutation of the first row. product [14,15], which captures connectivity characteristics that are less regular and therefore more heterogeneous than those found in the Cartesian product. Several graph product operators have been proposed and studied in mathematics, which di er from each other regarding how to connect those nodes in the product graph. (1) The complete p -partite graph K p a ( p > 1, a > 1) has clique number p and eigenvalues ( p 1) a, 0, a, where the multiplicity of 0 is p ( a 1). The energy of K p a is 2 ( p 1) a. Corollary 2.4 Let H1 be a graph and (,) an eigenpair for its adjacency matrix; let H2 be a graph and (,) an eigenpair for its adjacency matrix. For a regular space structure, the visualization of its graph model as the product of two simple graphs results in a substantial simplification in the solution of the corresponding eigenproblems. The only non trivial eigenvalue of the complete graph is nG(with multiplicity nG 1) and condition (13) yields q < nG < q+. a basic text in graph theory, it contains, for the rst time, Diracs theorem on k-connected graphs (with adequate hints), HararyNashWilliams theorem on the hamiltonicity of line graphs, ToidaMcKees characterization of Eulerian graphs, the Tutte matrix of a graph, David Sumners result on claw-free graphs, Fourniers We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0,) . Suppose G;Hare graphs with no loops. PROBLEM Find the eigenvalues of the graph obtains by removing ndisjoint edges from K 2n: 5. Leslie Hogben, Spectral graph theory and the inverse eigenvalue problem of a graph , The Electronic Journal of Linear Algebra: Vol. In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. Graphs and Eigenvalues Ho man Graphs (Ho man) Graphs with given smallest eigenvalue Limit points On (Ho man) graphs with smallest eigenvalue at Now consider the inner product (c x;c y). the eigenvalues of signed graph . Abstract Eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization. Introducing a coupling parameter describing the relative The cartesian product of \(2\) non-empty sets \(A\) and \(B\) is the set of all possible ordered pairs where the first component is from \(A\) and the second component is from \(B.\) I need to calculate the second-largest eigenvalue of the adjacency matrix. The union and join operations are dened Introducing a coupling parameter describing the relative the cartesian product of graphs; the decomposition of vertex set and the directed sum of graphs as binary or k-ary operations. Ren Descartes, a French mathematician and philosopher has coined the term Cartesian. In this paper we obtain the D-spectrum of the cartesian product if 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. GRAPHS AND CARTESIAN PRODUCTS OF COMPLETE GRAPHS BRIAN JACOBSON, ANDREW NIEDERMAIER, AND VICTOR REINER Abstract. 502 Eigenvalues of regular graphs. Multiplex networks are also obtained under specific prescriptions. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. This class of graphs have a close relationship to strongly regular graphs. 1 Answer Sorted by: 3 The grid graph is the Cartesian product of two copies of the path P n . For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. 139 Separating sets. A design graph is a regular bipartite graph in which any two distinct vertices of the same part have the same number of common neighbors. In this paper, we study the distance eigenvalues of the design graphs. This is proved using the same eigenvectors vfs as above (see As the main result, we use tensor products to prove a relation between the eigenvalues of the cartesian product of graphs and the eigenvalues of the original graphs. Denote the eigenvalues of a matrix M of order n by j (M) for j = 1, 2, . For graphs, there are a variety of different kinds of graph products: cartesian product, lexicographic (or ordered) product, tensor product, and strong product are The D-eigenvalues 1, 2, , p of a graph G are the eigenvalues of its distance matrix D and form the distance spectrum or the D-spectrum. The total graph is built by joining the graph to its line graph by means of the incidences. This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. The set of eigenvalues (with their multiplicities) of a graph G is the spectrum of its adjacency matrix and it is the spectrum of G and denoted by Sp (G). The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Dene graph G Hwhere V(G H) = f(g;h) : g2V(G);h2V(H)g; expressed as the graph Cartesian product of smaller sub-graphs, it admits a solution in linear time, thus, allowing to scale up to larger and more practical problems. The Cartesian product of and , written as , is the graph with vertex set , and two vertices and are adjacent whenever and or and . . 1. The energy of K n 1 K n 2 is 4 ( n 1 1 ) ( n 2 1 ) . It is known that a graph G is bipartite if and only if there is an orientation of G such that SpS(G)=iSp(G). reasonable estimation with percentage errors con ned within a 10% range for most eigenvalues. We study the distributions of edges crossed by a cut in G^k across the copies of G in different Two nodes (g;h)and (g;h)are connected in GjHif and only if The collection of eigenvalues of A(G) A ( G) together with multiplicities is called the A A -\emph {spectrum} of G G. Let G H G H, G[H] G [ H], GH G H and GH G H be the Cartesian product, lexicographic product, directed product and strong product of graphs G G and H H, respectively. In this paper, we focus on the following three fundamental graph products [5]: Cartesian product: Denoted as GjH. (This one is di cult). eorem . In this paper an efficient algorithm is presented for identifying the generators of regular graph models G formed by Cartesian graph products. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues. Moreover, in Section 4 we construct a scale free graph with = 1 with a small spectrum (only three positive eigenvalues). Abstract: The k-fold Cartesian product of a graph G is defined as a graph on k-tuples of vertices, where two tuples are connected if they form an edge in one of the positions and are equal in the rest. 14 Some Applications of Eigenvalues of Graphs 361 Theorem 3 (Matrix-Tree Theorem). Let C be the adjacency matrix for the Cartesian product H1 H2. Cartesian product and the corona product of signed graphs. In summary, a Cartesian product of n graphs is an n-dimensional graph whose each dimension is formed by each factor graph (its definition will be introduced in Section II-C). A graph Gwhose Laplacian matrix has integer eigenvalues is called Laplacian integral. Isometric embedding in cartesian products. In mathematics, multipli- for Cartesian product graphs. with Vizings conjecture on the domination number of the Cartesian product of two graphs. This can be generalized to Paley tournaments on q vertices, where q 3 mod 4. In other words, the number of nodes of G, or equivalently the number of layers in the multiplex, can act as a control parameter to instigate, or alternatively dissolve, the Turing instability. Given that 1, , n and 1, m are the eigenvalues of the Laplacians of G and H respectively, it is well known that the eigenvalues of the carteisan product of G and H are. The sign of a cycle in a signed graph is the product of the signs of its edges. In the meantime, there are other important forms of graph products, such as

Let 1; 2;:::; n be eigenvalues of A. A graph can be considered to be a homogeneous signed graph; thus signed graphs become a generalization of graphs. We start with some basic definitions in graph theory: incidence matrix, eigenvalues and cartesian product. One of the best known examples is the hypercube or n-cube, which can be seen as the cartesian (or direct) product of complete graphs on two vertices. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs The cartesian product affects eigenvalues in a similar way.