Eigenvalues can be used to nd the trace of a matrix raised to a power. The cartesian product affects eigenvalues in a similar way.

Recent work has used variations of the hypergraph eigenvalues we describe to obtain results about the maximal cliques in a hypergraph [6], cliques in 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. The dispersion relation which controls the onset of the instability depends on a set of discrete wavelengths, the eigenvalues of the aforementioned Laplacians.

Let G H be the Cartesian Product of G and H. Determine L ( G H) in terms of L ( G) and L ( H) where L ( G) denotes Laplacian Matrix of G. Also find the eigen values of L ( G H) in terms of L ( G) and L ( H). Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. : Expander graphs and coding theory.

Dene graph G Hwhere V(G H) = f(g;h) : g2V(G);h2V(H)g; We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor signed graphs. As for A , all the eigenvalues of L are real.

For any eigenvalue of Aand any eigenvalue of B, we would like to show + is an eigenvalue of G H.

Among all eigenvalues of the Laplacian of a graph, one of the most popular is the second smallest, called by Fiedler [25], the algebraic connectivity of a graph. Classical graphs can also display a modular or hierarchical structure. 129 Exercises.

0(a) shows an example of a two-dimensional a 2k-regular \k-dimensional grid graph," and only a weak expander for k xed and number of vertices large; (3)the Boolean hypercube; (4)more generally, the cartesian product G 1 G 2 of any two graphs in terms of the eigenvalues/vectors of G 1 and G 2; (5)other products; (6)Cayley graphs of abelian groups and (some remarks) about non-abelian groups. Some classes of Laplacian integral graphs have been identi ed.

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. is a generalization of the Cartesian product of ordinary graphs (see [6, Section 2.5]). The eigenvalues of the Laplacian of the Cartesian product of two graphs are the sum of the eigenvalues of the Laplacians of the graphs. This class of graphs have a close relationship to strongly regular graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. Relation Recall that the Cartesian product of two sets A and B, denoted A B, is the set of all ordered pairs (a, b) where a , and b B. Then the eigenvalues of A are given by 2 [ ] (1) , Irr (G), 1 where = (1) sS (s). 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. Also, we will explicitly determine the distance eigenvalues of a class of design graphs, and GRAPHS AND CARTESIAN PRODUCTS OF COMPLETE GRAPHS BRIAN JACOBSON, ANDREW NIEDERMAIER, AND VICTOR REINER Abstract. 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. 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. 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.

Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs.

Recommended papers. 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).

The sign of a cycle in a signed graph is the product of the signs of its edges. A signed graph is said [3, 13]. If Spec(G) = (Ai,, Am) and Spec(H) = (py,, pn), then Spec(GDi/) consists of all mn sums {Ar + ps: 1 < r < m, 1 < s < n}. Let C be the adjacency matrix for the Cartesian product H1 H2. We now dene graphproducts.Denote a generalgraphproductof twosimplegraphs by G H: We dene the product in such a way that G H is also simple. Given graphsG 1and G 2with vertexsets V 1and V 2respectively,any productgraphG 1G 2 has as its vertex set the Cartesian product V.G 1/ V.G 2/: For any two vertices .u 1;u 2/; .v 1;v 2/ of G 1G distribution with a regular graph is a scale free graph without eigenvalue power law distribution.

Consider a two-dimensional grid with wrap-around edges (a doughnut-shaped graph). The eigenvalue of A is said to be a main eigenvalue of G if the eigenspace E() is not orthogonal to the all-1 vector j. In particular, we will examine algorithms for solving linear systems and quantum algorithms. In the meantime, there are other important forms of graph products, such as

Eigenvalues of Cartesian Products Yiwei Fu 1.6 Eigenvalues of Cartesian Products Denition 1.6.1. A graph Gwhose Laplacian matrix has integer eigenvalues is called Laplacian integral.

Denote the eigenvalues of a matrix M of order n by j (M) for j = 1, 2, . We start with some basic definitions in graph theory: incidence matrix, eigenvalues and cartesian product. The eigenvalues of the adjacency matrix of a graph are often referred to as the eigenvalues of the graph and those of the Laplacian matrix as the Laplacian eigenvalues. 119 Product dimension. Then we introduce the tensor product of vector spaces. A graph can be considered to be a homogeneous signed graph; thus signed graphs become a generalization of graphs. The Cartesian product 1 2 of two signed graphs 1 = (V 1 , E 1 , 1 ) and 2 = (V 2 , E 2 , 2 ) is a generalization of the Cartesian product of ordinary graphs (see [6, Section 2.5]). Then + is an eigenvalue with eigenvector for C. Proof: Since m = 2, Theorem 2.3 implies m2 u = m2 x =1. with Vizings conjecture on the domination number of the Cartesian product of two graphs.

In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. In mathematics, multipli- for Cartesian product graphs. We introduce a similar construction for signed graphs.

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. In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. 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. The Cartesian product 1 2 of two signed graphs 1 = (V 1 , E 1 , 1 ) and 2 = (V 2 , E 2 , 2 ) is a generalization of the Cartesian product of ordinary graphs (see [6, Section 2.5]). 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). Introducing a coupling parameter describing the

This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates.

I need to calculate the second-largest eigenvalue of the adjacency matrix. 1 Introduction Calculating a product of multiple graphs has been studied in several disciplines. including disjoint unions, Cartesian products, k-partite graphs, k-cylinders, a generalization of the hypercube, and complete hypergraphs. mare eigenvalues of the adjacency matrix of a graph H. Then the eigenvalues of the adjacency matrix of the Cartesian product G H are i+ jfor 1 i nand 1 j m. Proof: Let A(or B) be the adjacency matrix of G(or H) respectively. 504 Strongly regular graphs. For two disjoint graphs and , the strong product of them is written as , that is, , and two distinct vertices and are contiguous. It is also well-known [9, Lemma 13.1.3] that if Ghas no multiple We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0,) . The critical group of a connected graph is a nite abelian group, and hence its eigenvalues are the (multiset) union of the eigenvalues for each Gi. It is also well-known [9, Lemma 13.1.3] that if Ghas no multiple This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. A signed graph is said [3, 13]. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended p-sum, or NEPS) of signed graphs. 139 Eigenvalues and graph parameters. If 1 and 2 are the regular graph of degrees - and , respectively, the eigenvalues of the Kirchho matrix (1) are written as 0= 0 1 1,andthe eigenvalues of the Kirchho matrix (2) are written as 0= F0 F1 F 1, then the number of spanning trees of the Cartesian product of 1 502 Eigenvalues of regular graphs. Key W ords: Signed graph, Cartesian pro duct graph, Line graph, Graph Laplacian, Kirchho matrix, Eigenv alues of graphs, Energy of graphs.

The kth eigenvalue of K, is n-1 ifk=O -1 if k f 0, Ak= where the eigenvalue -1 has multiplicity n - 1.

A graph is called prime if it cannot be decomposed into the product of non-trivial graphs, otherwise a graph is referred to as composite. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and In this seminar, we will explore and exploit eigenvalues and eigenvectors of graphs. The energy of K n 1 K n 2 is 4 ( n 1 1 ) ( n 2 1 ) . 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.

. spectrum SpecG of G is the set of eigenvalues of A G. The graph G is called integral if all of its eigenvalues are integers. Moreover, in Section 4 we construct a scale free graph with = 1 with a small spectrum (only three positive eigenvalues).

When raising the adjacency matrix to a power the entries count the number of closed walks. The word Cartesian product is made of two words, i.e., Cartesian and product. 1 Answer Sorted by: 3 The grid graph is the Cartesian product of two copies of the path P n . 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.

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. as it has

In 1978, I. Gutman introduced the concept of energy of a graph [4], the energy of Gis dened as E(G) = 3 and the cartesian product graph K 2 C 3 with V(K 2 C 3) = fw 1;w 2;w 3;w 4;w 5;w 6g:and A(K 2 C 3) be its adjacency matrix. 1.

(2020) by means of different constructions.

the eigenvalues of signed graph . Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Let 1; 2;:::; n be eigenvalues of A.

We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs. the cartesian product of graphs; the decomposition of vertex set and the directed sum of graphs as binary or k-ary operations. Their dot product is 2*-1 + 1*2 = 0.

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. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Abstract Eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization. It can be shown that matrix L is a positive semidefinite matrix with 10 and 2.4.

and is the set of all eigenvalues of Gwith their multiplicity. Multiplex networks are also obtained under specific prescriptions. Leslie Hogben, Spectral graph theory and the inverse eigenvalue problem of a graph , The Electronic Journal of Linear Algebra: Vol.