Tensor Product Of Graphs Research Articles

A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

A kite K is a graph which can be obtained by joining an edge to any vertex of K 3 . A kite with edge set { a b , b c , c a , c d } can be denoted as ( a , b , c ; c d ) . If every vertex of a kite in the decomposition lies in different partite sets, then we say that a kite decomposition of a multipartite graph is a gregarious kite decomposition. In this manuscript, it is shown that there exists a decomposition of ( K m ⊗ ¯¯¯¯¯ K n ) × ( K r ⊗ ¯¯¯¯¯ K s ) into gregarious kites if and only if n 2 s 2 m ( m − 1 ) r ( r − 1 ) ≡ 0 ( mod 8 ) , where ⊗ and × denote the wreath product and tensor product of graphs respectively. We denote a gregarious kite decomposition as G K -decomposition.

Scalable Label Propagation for Multi-Relational Learning on the Tensor Product of Graphs

Multi-relational learning on knowledge graphs infers high-order relations among the entities across the graphs. This learning task can be solved by label propagation on the tensor product of the knowledge graphs to learn the high-order relations as a tensor. In this paper, we generalize a widely used label propagation model to the normalized tensor product graph, and propose an optimization formulation and the scalable Low-rank Tensor-based Label Propagation algorithm (LowrankTLP) to infer multi-relations for two learning tasks, hyperlink prediction and multiple graph alignment. The optimization formulation minimizes the upper bound of the noisy-tensor estimating error for multiple graph alignment, by learning with a subset of the eigen-pairs in the spectrum of the normalized tensor product graph. We also provide a data-dependent transductive Rademacher bound for binary hyperlink prediction. We accelerate LowrankTLP with parallel tensor computation which enables label propagation on a tensor product of 100 graphs each of size 1000 in less than half hour in the simulation. LowrankTLP was also applied to predicting the author-paper-venue hyperlinks in publication records, alignment of segmented regions across up to 26 CT-scan images and alignment of protein-protein interaction networks across multiple species. The experiments demonstrate that LowrankTLP indeed well approximates the original label propagation with better scalability and accuracy.

Clustering Coefficient of the Tensor Product of Graphs

Clustering coefficient is one of the most useful indices in complex networks. However, graph theoretic properties of this metric have not been discussed much in the literature, especially in graphs resulting from some binary operations. In this paper we present some expressions for the clustering coefficient of the tensor product of arbitrary graphs, regular graphs, and strongly regular graphs. A Vizing-type upperbound and a sharp lower bound for the clustering coefficient of the tensor product of graphs are also given.

EGHWT: The Extended Generalized Haar\u2013Walsh Transform

Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important, and much research has been done recently. The generalized Haar–Walsh transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh–Hadamard transforms. We propose the extended generalized Haar–Walsh transform (eGHWT), which is a generalization of the adapted time–frequency tilings of Thiele and Villemoes (1996). The eGHWT examines not only the efficiency of graph-domain partitions but also that of “sequency-domain” partitions simultaneously. Consequently, the eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost, O(N log N), where N is the number of nodes of an input graph. While the GHWT best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than (1.5)^N possible orthonormal bases in mathbb {R}^N, the eGHWT best-basis algorithm can find a better one by searching through more than 0.618cdot (1.84)^N possible orthonormal bases in mathbb {R}^N. This article describes the details of the eGHWT best-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Furthermore, we also show how the eGHWT can be extended to 2D signals and matrix-form data by viewing them as a tensor product of graphs generated from their columns and rows and demonstrate its effectiveness on applications such as image approximation.

Construction of Mutually Orthogonal Graph Squares Using Novel Product Techniques

Sets of mutually orthogonal Latin squares prescribe the order in which to apply different treatments in designing an experiment to permit effective statistical analysis of results, they encode the incidence structure of finite geometries, they encapsulate the structure of finite groups and more general algebraic objects known as quasigroups, and they produce optimal density error‐correcting codes. This paper gives some new results on mutually orthogonal graph squares. Mutually orthogonal graph squares generalize orthogonal Latin squares interestingly. Mutually orthogonal graph squares are an area of combinatorial design theory that has many applications in optical communications, wireless communications, cryptography, storage system design, algorithm design and analysis, and communication protocols, to mention just a few areas. In this paper, novel product techniques of mutually orthogonal graph squares are considered. Proposed product techniques are the half‐starters’ vectors Cartesian product, half‐starters’ function product, and tensor product of graphs. It is shown that by taking mutually orthogonal subgraphs of complete bipartite graphs, one can obtain enough mutually orthogonal subgraphs in some larger complete bipartite graphs. Also, we try to find the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed different product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed. All the constructed results in this paper can be used to generate new graph‐orthogonal arrays and new authentication codes.

Inverse Majority Neighborhood Number for Tensor Product of Graphs

For the tensor product of paths, inverse majority neighborhood number n m −1 (G) of G are determined in this paper. Also obtained some bounds for G = P n ⊗ P m .

Decompositions of Some Classes of Dense Graphs into Cycles of Lengths 4 and 8

For an even graph G and non-negative integers r and s, the pair $$(r,\,s)$$ is an admissible pair if $$4r+8s=|E(G)|.$$ If G admits a decomposition into r copies of $$C_4,$$ the cycle of length four, and s copies of $$C_8,$$ the cycle of length eight, for every admissible pair $$(r,\,s),$$ then G has a $$\{C_4^r,\,C_8^s\}$$ -decomposition. In this paper, a necessary and sufficient condition is obtained for the existence of a $$\{C_4^r,\,C_8^s\}$$ -decomposition of the complete k-partite graph $$K_{a_1,\,a_2,\,\dots ,\,a_k},$$ where $$k\ge 3.$$ Further, a characterization is obtained for the graph $$K_m\times K_n,$$ where $$\times$$ denotes the tensor product of graphs, to admit a $$\{C_4^r,\,C_8^s\}$$ -decomposition.

T-Coloring of product graphs

Given a graph [Formula: see text] and a finite set [Formula: see text] of positive integers containing [Formula: see text] a [Formula: see text]-coloring of [Formula: see text] is a function [Formula: see text] for all [Formula: see text] in [Formula: see text] such that if [Formula: see text] then [Formula: see text] For a [Formula: see text]-coloring [Formula: see text] of [Formula: see text], the [Formula: see text]-span [Formula: see text] is the maximum value of [Formula: see text] over all pairs [Formula: see text] of vertices of [Formula: see text] The [Formula: see text]-span [Formula: see text] is the minimum [Formula: see text]-span overall [Formula: see text]-colorings of [Formula: see text]. The [Formula: see text]-edge span of a [Formula: see text]-coloring [Formula: see text] is the maximum value of [Formula: see text] over all edges [Formula: see text] of [Formula: see text]. The [Formula: see text]-edge span [Formula: see text] is the minimum [Formula: see text]-edge span overall [Formula: see text]-colorings of [Formula: see text]. This paper discusses the [Formula: see text]-span and [Formula: see text]-edge span of Cartesian, Join, Union and Tensor products of graphs. Also, we discuss the relation between Restricted span and Restricted edge span of graphs.

Multitype branching and graph product theory of infectious disease outbreaks.

The heterogeneity of human populations is a challenge to mathematical descriptions of epidemic outbreaks. Numerical simulations are deployed to account for the many factors influencing the spreading dynamics. Yet, the results from numerical simulations are often as complicated as the reality, leaving us with a sense of confusion about how the different factors account for the simulation results. Here, using a multitype branching together with a graph tensor product approach, I derive a single equation for the effective reproductive number of an infectious disease outbreak. Using this equation I deconvolute the impact of crowd management, targeted testing, contact heterogeneity, stratified vaccination, mask use, and smartphone tracing app use. This equation can be used to gain a basic understanding of infectious disease outbreaks and their simulations.

On (2-d)-Kernels in the Tensor Product of Graphs

In this paper, we study the existence, construction and number of (2-d)-kernels in the tensor product of paths, cycles and complete graphs. The symmetric distribution of (2-d)-kernels in these products helps us to characterize them. Among others, we show that the existence of (2-d)-kernels in the tensor product does not require the existence of a (2-d)-kernel in their factors. Moreover, we determine the number of (2-d)-kernels in the tensor product of certain factors using Padovan and Perrin numbers.

The Tensor Product of Zero-Divisor Graphs of Variation Monogenic Semigroups

The tensor product of zero-divisor graphs of variation monogenic semigroups Γ(VS_Mn^1) and Γ(VS_Mm^2) is studied. The vertices(x_1^i,x_2^j) and (x_1^k,x_2^f) of the tensor product of this graph are adjacent whenever gcd(i,k)=1,i+k>n,gcd(j,f)=1 ,j+f>m. Some properties of tensor product graphs are obtained, such as girth, diameter, chromatic, clique and domination numbers.

Equitable edge coloring on tensor product of graphs

A graph G is edge colored if different colors are assigned to its edges or lines, in the order of neighboring edges are allotted with least diverse k-colors. If each of k-colors can be partitioned into color sets and differs by utmost one, then it is equitable. The minimum of k-colors required is known as equitably edge chromatic number and symbolized by $\chi^{\prime}_{=}(G)$. Further the impression of equitable edge coloring was first initiated by Hilton and de Werra in 1994. In this paper, we ascertain the equitable edge chromatic number of $P_m \otimes P_n$, $P_m \otimes C_n$ and $K_{1,m} \otimes K_{1,n}$.

High performance GPU primitives for graph-tensor learning operations

Graph-tensor learning operations extend tensor operations by taking the graph structure into account, which have been applied to diverse domains such as image processing and machine learning. However, the running time of graph-tensor operations increases rapidly with the number of nodes and the dimension of data on nodes, making them impractical for real-time applications. In this paper, we propose a GPU library called cuGraph-Tensor for high-performance graph-tensor learning operations, which consists of eight key operations: graph shift (g-shift), graph Fourier transform (g-FT), inverse graph Fourier transform (inverse g-FT), graph filter (g-filter), graph convolution (g-convolution), graph-tensor product (g-product), graph-tensor SVD (g-SVD) and graph-tensor QR (g-QR). cuGraph-Tensor supports scalar, vector, and matrix data processing on each graph node. We propose optimization techniques on computing, memory accesses, and CPU–GPU communications that significantly improve the performance of the graph-tensor learning operations. Using the optimized operations, cuGraph-Tensor builds a graph data completion application for fast and accurate reconstruction of incomplete graph data. In the experiments, the proposed graph learning operations achieve up to 142.12× speedups versus CPU-based GSPBOX and CPU MATLAB implementations running on two Xeon CPUs. The graph data completion application achieves up to 174.38× speedups over the CPU MATLAB implementation, and up to 3.82× speedups with better accuracy over the GPU-based tensor completion in the cuTensor-tubal library.

\(C_{6}\)-decompositions of the tensor product of complete graphs

Let \(G\) be a simple and finite graph. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). If \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\), where \(H_1\), \(H_2\), ..., \(H_k\) are all isomorphic to \(H\), then \(G\) is said to be \(H\)-decomposable. Furthermore, if \(H\) is a cycle of length \(m\) then we say that \(G\) is \(C_m\)-decomposable and this can be written as \(C_m|G\). Where \( G\times H\) denotes the tensor product of graphs \(G\) and \(H\), in this paper, we prove that the necessary conditions for the existence of \(C_6\)-decomposition of \(K_m \times K_n\) are sufficient. Using these conditions it can be shown that every even regular complete multipartite graph \(G\) is \(C_6\)-decomposable if the number of edges of \(G\) is divisible by \(6\).

Some classifications of graphs with respect to a set adjacency relation

For any finite simple undirected graph [Formula: see text], we consider the binary relation [Formula: see text] on the powerset [Formula: see text] of its vertex set given by [Formula: see text] if [Formula: see text], where [Formula: see text] denotes the neighborhood of a vertex [Formula: see text]. We call the above relation set adiacence dependency (sa)-dependency of [Formula: see text]. With the relation [Formula: see text] we associate an intersection-closed family [Formula: see text] of vertex subsets and the corresponding induced lattice [Formula: see text], which we call sa-lattice of [Formula: see text]. Through the equality of sa-lattices, we introduce an equivalence relation [Formula: see text] between graphs and propose three different classifications of graphs based on such a relation. Furthermore, we determine the sa-lattice for various graph families, such as complete graphs, complete bipartite graphs, cycles and paths and, next, we study such a lattice in relation to the Cartesian and the tensor product of graphs, verifying that in most cases it is a graded lattice. Finally, we provide two algorithms, namely, the T-DI ALGORITHM and the O-F ALGORITHM, in order to provide two different computational ways to construct the sa-lattice of a graph. For the O-F ALGORITHM we also determine its computational complexity.

The first general Zagreb coindex of graph operations

Abstract Many chemically important graphs can be obtained from simpler graphs by applying different graph operations. Graph operations such as union, sum, Cartesian product, composition and tensor product of graphs are among the important ones. In this paper, we introduce a new invariant which is named as the first general Zagreb coindex and defined as M ¯ 1 α ( G ) = ∑ uv ∈ E ( G ¯ ) [ d G ( u ) α + d G ( v ) α ] \overline{M}^\alpha_1(G)=\Sigma_{uv\in E(\overline{G})}[d_G(u)^\alpha+d_G(v)^\alpha] , where α ∈ ℝ, α ≠ 0. Here, we study the basic properties of the newly introduced invariant and its behavior under some graph operations such as union, sum, Cartesian product, composition and tensor product of graphs and hence apply the results to find the first general Zagreb coindex of different important nano-structures and molecular graphs.

$C_4$-decomposition of the tensor product of complete graphs

Let G be a simple and finite graph. A graph is said to be decomposed into subgraphs H 1 and H 2 which is denoted by G = H 1 ⊕ H 2 , if G is the edge disjoint union of H 1 and H 2 . If G = H 1 ⊕ H 2 ⊕ H 3 ⊕ ... ⊕ H k , where H 1 , H 2 , H 3 , ..., H k are all isomorphic to H , then G is said to be H -decomposable. Futhermore, if H is a cycle of length m then we say that G is C m -decomposable and this can be written as C m | G . Where G × H denotes the tensor product of graphs G and H , in this paper, we prove the necessary and sufficient conditions for the existence of C 4 -decomposition of K m × K n . Using these conditions it can be shown that every even regular complete multipartite graph G is C 4 - decomposable if the number of edges of G is divisible by 4.

FOUR NEW TENSOR PRODUCTS OF GRAPHS AND THEIR ZAGREB INDICES AND COINDICES B. BASAVANAGOUD, V. R. DESAI, KEERTHI G. MIRAJKAR, B. POOJA AND ISMAIL NACI CANGUL

FOUR NEW TENSOR PRODUCTS OF GRAPHS AND THEIR ZAGREB INDICES AND COINDICES B. BASAVANAGOUD, V. R. DESAI, KEERTHI G. MIRAJKAR, B. POOJA AND ISMAIL NACI CANGUL

On star coloring of tensor product of graphs

On star coloring of tensor product of graphs

Far East Journal of Mathematical Sciences (FJMS) Volume 118, Issue 1, Pages 77 - 105 (September 2019) http://dx.doi.org/10.17654/MS118010077 ADJACENT VERTEX DISTINGUISHING TOTAL COLORING OF TENSOR PRODUCT OF GRAPHS

Far East Journal of Mathematical Sciences (FJMS) Volume 118, Issue 1, Pages 77 - 105 (September 2019) http://dx.doi.org/10.17654/MS118010077 ADJACENT VERTEX DISTINGUISHING TOTAL COLORING OF TENSOR PRODUCT OF GRAPHS