# eigenvalues of cartesian product of graphs

: Expander graphs and coding theory. 504 Strongly regular graphs.

We start with some basic definitions in graph theory: incidence matrix, eigenvalues and cartesian product. In this paper, we study the distance eigenvalues of the design graphs. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs Isometric embedding in cartesian products.

In this paper, we study the edge tenacity of graphs. A signed graph is said [3, 13]. The only real eigenvalue is 3, the remaining eigenvalues are equal to ( 1 7) / 2, with absolute value 2. 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. including disjoint unions, Cartesian products, k-partite graphs, k-cylinders, a generalization of the hypercube, and complete hypergraphs. 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.

The Cartesian product of and , written as , is the graph with vertex set , and two vertices and are adjacent whenever and or and . 1. It is also well-known [9, Lemma 13.1.3] that if Ghas no multiple 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. 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.

Abstract Eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization. Also, we will explicitly determine the distance eigenvalues of a class of design graphs, and The eigenvalue based methods have proved to be useful also for some other problems, e.g. This class of graphs have a close relationship to strongly regular graphs. The only non trivial eigenvalue of the complete graph is nG(with multiplicity nG 1) and condition (13) yields q < nG < q+. PROBLEM Find the eigenvalues of the graph obtains by removing ndisjoint edges from K 2n: 5. the eigenvalues of signed graph . Moreover, in Section 4 we construct a scale free graph with = 1 with a small spectrum (only three positive eigenvalues). I need to calculate the second-largest eigenvalue of the adjacency matrix.

Recent work has used variations of the hypergraph eigenvalues we describe to obtain results about the maximal cliques in a hypergraph [6], cliques in We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0,) . Denote the eigenvalues of a matrix M of order n by j (M) for j = 1, 2, .

Energy of a graph, equienergetic graphs, Cartesian product, generalized composition, equitable partition. In this paper, we characterize the extremal graphs attaining the upper bound n 2 and the second upper bound n 3.

the cartesian product of graphs; the decomposition of vertex set and the directed sum of graphs as binary or k-ary operations. 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]). As for A , all the eigenvalues of L are real. Dene graph G Hwhere V(G H) = f(g;h) : g2V(G);h2V(H)g; A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators.

We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and Let 1; 2;:::; n be eigenvalues of A. is a generalization of the Cartesian product of ordinary graphs (see [6, Section 2.5]).

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. When raising the adjacency matrix to a power the entries count the number of closed walks.

Cartesian product and the corona product of signed graphs. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, The eigenvalues of the Laplacian of the Cartesian product of two graphs are the sum of the eigenvalues of the Laplacians of the graphs.

For example, let B be a set of blouses and S be a set of skirts. We study the distributions of edges crossed by a cut in G^k across the copies of G in different Then + is an eigenvalue with eigenvector for C. Proof: Since m = 2, Theorem 2.3 implies m2 u = m2 x =1. It can be shown that matrix L is a positive semidefinite matrix with 10 and 2.4. We introduce a similar construction for signed graphs. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues. If A is a square matrix, the eigenvalues are the scalar values u satisfying Ax = ux, and the eigenvectors are the values of x. Eigenvectors and eigenvalues give a convenient representation of matrices for computing powers of matrices and for solving differential equations.

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. Starting with G as a single edge gives G^k as a k-dimensional hypercube. i + j for i = 1, , n and j = 1, , m. Thus a cycle is positive if and only if it contains an even number of negative edges.

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.

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 Some graph operations such as the Cartesian product and the strong product may be used to generate new integral graphs from given ones [8]. The cartesian product affects eigenvalues in a similar way. The union and join operations are dened Let G be a (nite, undirected, simple) graph with node set V(G) = f1;:::;ng. (This one is di cult).

Explore the eigenvalues and eigenvectors of G Hfor two graphs Gand H. In particular, consider Q nwhich is the n-fold cartesian product of P 2. . Introduction A signed graph is a pair ( , ), where = ( , ) is a simple unsigned graph,

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. Let C be the adjacency matrix for the Cartesian product H1 H2. 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.

Ren Descartes, a French mathematician and philosopher has coined the term Cartesian. It is also well-known [9, Lemma 13.1.3] that if Ghas no multiple

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. Then the eigenvalues of A are given by 2 [ ] (1) , Irr (G), 1 where = (1) sS (s). Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. This study focuses on signals on a Cartesian product of graphs, which are termed multi-dimensional graph signals hereafter. (2020) by means of different constructions. as it has 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.

The adjacency 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.

14 Some Applications of Eigenvalues of Graphs 361 Theorem 3 (Matrix-Tree Theorem). Graph products 2.4.1. The Cartesian product G x H of graphs G and H Thus a cycle is positive if and only if it contains an even number of negative edges. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted 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. This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates. 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. For two disjoint graphs and , the strong product of them is written as , that is, , and two distinct vertices and are contiguous.

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 The eigenvalues of the line signed graph (+G) of G with all positive signs are 2 L 1 (+G),,2 L nc (G) (+G)<2 and eigenvalue 2 with multiplicity m n+ c (G).

A graph Gwhose Laplacian matrix has integer eigenvalues is called Laplacian integral.

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. The total graph is built by joining the graph to its line graph by means of the incidences. ABSTRACT.

Interest- and is a diagonal matrix with eigenvalues on the diagonal. 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. , n. 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.

Derive an Alon-Boppana type bound for non-regular graphs? GRAPHS AND CARTESIAN PRODUCTS OF COMPLETE GRAPHS BRIAN JACOBSON, ANDREW NIEDERMAIER, AND VICTOR REINER Abstract. 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 1. 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. 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.

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. Let G be a finite connected graph on two or more vertices, and G [N,k] the distance-k graph of the N-fold Cartesian power of G. For a fixed k 1, we obtain explicitly the large N limit of the spectral distribution (the eigenvalue distribution of the adjacency matrix) of G [N,k].The limit distribution is described in terms of the Hermite polynomials. The same kind of problem has been addressed to the eigenvalues of the Laplacian matrix.

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 Eigenvalues can be used to nd the trace of a matrix raised to a power. Filomat 9, 449472 (1995) MathSciNet MATH Google Scholar Dowling Jr, M.C.

The characteristic polynomial of the adjacency matrix is ( x 3) ( x 2 + x + 2) 3. GRAPHS AND CARTESIAN PRODUCTS OF COMPLETE GRAPHS BRIAN JACOBSON, ANDREW NIEDERMAIER, AND VICTOR REINER Abstract. is a generalization of the Cartesian product of ordinary graphs (see [6, Section 2.5]). and is the set of all eigenvalues of Gwith their multiplicity. 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.

507 Laplacian eigenvalues.

Introducing a coupling parameter describing the relative 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. Cartesian product of two graphs. My try: Let | V ( G) | = We will study what eigenvalues and eigenvectors tell us about a graph, and see how this information may be used to design and analyze algorithms.

Classical graphs can also display a modular or hierarchical structure. .

For graphs, there are a variety of different kinds of graph products: cartesian product, lexicographic (or ordered) product, tensor product, and strong product are Algebraic operations on graphs such as Cartesian product, Kronecker product, and direct sum can be used to generate new graphs from parent graphs.

In [D. Cui, Y. Hou, On the skew spectra of Cartesian products of

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.

Key W ords: Signed graph, Cartesian pro duct graph, Line graph, Graph Laplacian, Kirchho matrix, Eigenv alues of graphs, Energy of graphs.

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.

139 Eigenvalues and graph parameters. Patterns can develop on the Cartesian network, if they are supported on at least one of its constitutive sub-graphs. Leslie Hogben, Spectral graph theory and the inverse eigenvalue problem of a graph , The Electronic Journal of Linear Algebra: Vol. Consider the following vectors:. The connection between eigenvalues and cuts in graphs has been first discovered by Fiedler. 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).

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. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. Therefore, the entries of L are as: 8 Consider the following eigenproblem: 9 where i is the eigenvalue and v i is the corresponding eigenvector. We will be primarilyinterested in edge-tenacious graphs, which can be considered very stable and are somewhat analogous in edge tenacityto honest graphs in edge-integrity.

An eigenvector x is a main eigenvector if x>j 6= 0. Example: Orthogonality. This is 2 if x = y, 1 if x y and 0 otherwise. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. An example of a Cartesian product of two factor graphs is displayed in Figure 1.

2 1 + 2 2 + + 2n is the trace of A2 so is equal to twice the

Introducing a coupling parameter describing the mk , 2010 Mathematics Subject Classication.

The word Cartesian product is made of two words, i.e., Cartesian and product.

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. Recommended papers. Under two similar defnitions of the line signed graph, we defne the corresponding total signed graph and we show that it is stable under switching.

with Vizings conjecture on the domination number of the Cartesian product of two graphs. Let G be a simple graph with vertex set V(G) = {1,2,,n} and (0,1)-adjacency matrix A. eorem . The Cartesian product K n 1 K n 2 (n 1 n 2 2) has clique number n 1 and eigenvalues n 1 + n 2 2, n 1 2, n 2 2, 2 (the multiplicity of 2 is (n 1 1) (n 2 1)). Given a graph G, let G be an oriented graph of G with the orientation and skew-adjacency matrix S(G). 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. In this paper an efficient algorithm is presented for identifying the generators of regular graph models G formed by Cartesian graph products.

Consider a two-dimensional grid with wrap-around edges (a doughnut-shaped graph). not to complicate notation, well use the cross product in this case as well. The hypercube has been considered in parallel computers, (Ncube, iPSC/860, TMC CM-2, etc.) 119 Product dimension. 05C50.

Their dot product is 2*-1 + 1*2 = 0. 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.

A graph can be considered to be a homogeneous signed graph; thus signed graphs become a generalization of graphs. 502 Eigenvalues of regular graphs. Then, the kernel matrix could be expressed as follows: K = Ur 1() UT; (5) where r 1() = diag 1 r( i) . the eigenvalues of signed graph .

Some classes of Laplacian integral graphs have been identi ed.

It is known that (G) n 2 if G is a simple graph on n vertices and G is not isomorphic to nK1.

2 is that the Cartesian product of the path of length 2 and a complete graph has smallest eigenvalue 1 p For any eigenvalue of Aand any eigenvalue of B, we would like to show + is an eigenvalue of G H.

In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. On the hull sets and hull number of the cartesian product of graphs 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 spectrum of S(G) is called the skew-spectrum of G, denoted by SpS(G).

The kth eigenvalue of K, is n-1 ifk=O -1 if k f 0, Ak= where the eigenvalue -1 has multiplicity n - 1. Spectral graph theory Discrepancy Coverings Interlacing An application of the adjacency matrix. The main eigenvalues of the connected graphs of 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]).

In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. Introducing a coupling parameter describing the relative We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0,) . 1 Answer Sorted by: 3 The grid graph is the Cartesian product of two copies of the path P n . 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}.

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 this seminar, we will explore and exploit eigenvalues and eigenvectors of graphs. 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 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 $$B.$$

The second largest eigenvalue of a graph (a survey). This estimate is independent of the size of the graph and provides a general method to obtain higher order spectral estimates.

The sign of a cycle in a signed graph is the product of the signs of its edges. The dispersion relation which controls the onset of the instability depends on a set of discrete wavelengths, the eigenvalues of the aforementioned Laplacians. F.Harary and A.J.Schwenk [12]. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the 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 D-eigenvalues of the distribution with a regular graph is a scale free graph without eigenvalue power law distribution. If i j are two vertices of a connected graph G, then the number of spanning trees of G equals the absolute value of det.L.ij//.Also, the number of spanning trees ofG equals 2::: n n. We list now some simple properties of the eigenvalues of the Laplacian of a graph.