scholarly journals Adjacency Matrix of Product of Graphs

10.29007/fqlw ◽  
2018 ◽  
Author(s):  
Urvashi Acharya ◽  
Himali Mehta
Keyword(s):  
Graph Theory ◽  
Tensor Product ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
New Product ◽  
Different Types ◽  
Product Of Graphs

In graph theory, different types of matrices associated with graph, e.g. Adjacency matrix, Incidence matrix, Laplacian matrix etc. Among all adjacency matrix play an important role in graph theory. Many products of two graphs as well as its generalized form had been studied, e.g., cartesian product, 2−cartesian product, tensor product, 2−tensor product etc. In this paper, we discuss the adjacency matrix of two new product of graphs G H, where = ⊗2, ×2. Also, we obtain the spectrum of these products of graphs.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

scholarly journals Vertex Degrees and Isomorphic Properties in Complement of an m-Polar Fuzzy Graph

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Ch. Ramprasad ◽  
P. L. N. Varma ◽  
S. Satyanarayana ◽  
N. Srinivasarao
Keyword(s):  
Graph Theory ◽  
Tensor Product ◽  
Computer Science ◽  
Computational Intelligence ◽  
Cartesian Product ◽  
Combinatorial Problems ◽  
Fuzzy Graph ◽  
Normal Product ◽  
Product Graphs ◽  
Vertex Degrees

Computational intelligence and computer science rely on graph theory to solve combinatorial problems. Normal product and tensor product of an m-polar fuzzy graph have been introduced in this article. Degrees of vertices in various product graphs, like Cartesian product, composition, tensor product, and normal product, have been computed. Complement and μ-complement of an m-polar fuzzy graph are defined and some properties are studied. An application of an m-polar fuzzy graph is also presented in this article.

scholarly journals Interval-Valued Fuzzy Soft Graphs

2017 ◽  
Vol 5 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Onur Zihni ◽  
Yıldıray Çelik ◽  
Güven Kara
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Soft Sets ◽  
Strong Product ◽  
Different Types ◽  
Fuzzy Soft Sets ◽  
Interval Valued

Abstract In this paper, we combine concepts of interval-valued fuzzy soft sets and graph theory. Then we introduce notations of interval-valued fuzzy soft graphs and complete interval-valued fuzzy soft graphs. We also present several different types operations including cartesian product, strong product and composition on interval-valued fuzzy soft graphs and investigate some properties of them.

scholarly journals Study of Circular Distance in Graphs

2021 ◽  
Vol 12 (2) ◽  
pp. 2437-2444
Author(s):  
Peruri Lakshmi Narayana Varma , Et. al.
Keyword(s):  
Graph Theory ◽  
Significant Role ◽  
Geodesic Distance ◽  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Pendent Vertices ◽  
Product Of Graphs

Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph  and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph  . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖  P〗_2×C_n.  Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed. 

On bipolar fuzzy soft graphs

2018 ◽  
Vol 27 (2) ◽  
pp. 123-132
Author(s):  
Yildiray Çelik ◽  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Soft Sets ◽  
Strong Product ◽  
Different Types ◽  
Fuzzy Soft Sets

In this paper, we combine the concepts of bipolar fuzzy soft sets and graph theory. Then we introduce notations of bipolar fuzzy soft graph and strong bipolar fuzzy soft graph. We also present several different types of operations including cartesian product, strong product and composition on bipolar fuzzy soft graphs and investigate some properties of them.

Conjugate Laplacian matrices of a graph

2018 ◽  
Vol 10 (06) ◽  
pp. 1850082
Author(s):  
Somnath Paul
Keyword(s):  
Cartesian Product ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Discrete Mathematics ◽  
Laplacian Matrices ◽  
Cartesian Product Of Graphs ◽  
Laplacian Spectra ◽  
The Matrix ◽  
Matrix Formula ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph of order [Formula: see text] Let [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are two nonzero integers and [Formula: see text] is a positive integer such that [Formula: see text] is not a perfect square. In [M. Lepovi[Formula: see text], On conjugate adjacency matrices of a graph, Discrete Mathematics 307 (2007) 730–738], the author defined the matrix [Formula: see text] to be the conjugate adjacency matrix of [Formula: see text] if [Formula: see text] for any two adjacent vertices [Formula: see text] and [Formula: see text] for any two nonadjacent vertices [Formula: see text] and [Formula: see text] and [Formula: see text] if [Formula: see text] In this paper, we define conjugate Laplacian matrix of graphs and describe various properties of its eigenvalues and eigenspaces. We also discuss the conjugate Laplacian spectra for union, join and Cartesian product of graphs.

scholarly journals Spectral characterizations of sun graphs and broken sun graphs

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Romain Boulet
Keyword(s):  
Graph Theory ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Difficult Problem ◽  
Laplacian Spectrum ◽  
Pendant Vertex ◽  
Unicyclic Graphs ◽  
Adjacency Spectrum ◽  
International Audience ◽  
Spectral Characterizations

International audience Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns.

scholarly journals On the 1-2-3-conjecture

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Akbar Davoodi ◽  
Behnaz Omoomi
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Cartesian Product Of Graphs ◽  
Edge Weighting ◽  
International Audience ◽  
Product Of Graphs ◽  
Proper Vertex

Graph Theory International audience A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

scholarly journals The first general Zagreb coindex of graph operations

2020 ◽  
Vol 5 (2) ◽  
pp. 109-120
Author(s):  
Melaku Berhe Belay ◽  
Chunxiang Wang
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Molecular Graphs ◽  
Basic Properties ◽  
Graph Operations ◽  
Nano Structures ◽  
Product Composition ◽  
Product Of Graphs

AbstractMany 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 \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.

scholarly journals Identifying cell types from single-cell data based on similarities and dissimilarities between cells

2021 ◽  
Vol 22 (S3) ◽  
Author(s):  
Yuanyuan Li ◽  
Ping Luo ◽  
Yi Lu ◽  
Fang-Xiang Wu
Keyword(s):  
Gene Expression ◽  
Single Cell ◽  
Spectral Clustering ◽  
Incidence Matrix ◽  
Expression Patterns ◽  
Cell Types ◽  
Clustering Method ◽  
Different Types ◽  
Cell Data ◽  
Spectral Clustering Method

Abstract Background With the development of the technology of single-cell sequence, revealing homogeneity and heterogeneity between cells has become a new area of computational systems biology research. However, the clustering of cell types becomes more complex with the mutual penetration between different types of cells and the instability of gene expression. One way of overcoming this problem is to group similar, related single cells together by the means of various clustering analysis methods. Although some methods such as spectral clustering can do well in the identification of cell types, they only consider the similarities between cells and ignore the influence of dissimilarities on clustering results. This methodology may limit the performance of most of the conventional clustering algorithms for the identification of clusters, it needs to develop special methods for high-dimensional sparse categorical data. Results Inspired by the phenomenon that same type cells have similar gene expression patterns, but different types of cells evoke dissimilar gene expression patterns, we improve the existing spectral clustering method for clustering single-cell data that is based on both similarities and dissimilarities between cells. The method first measures the similarity/dissimilarity among cells, then constructs the incidence matrix by fusing similarity matrix with dissimilarity matrix, and, finally, uses the eigenvalues of the incidence matrix to perform dimensionality reduction and employs the K-means algorithm in the low dimensional space to achieve clustering. The proposed improved spectral clustering method is compared with the conventional spectral clustering method in recognizing cell types on several real single-cell RNA-seq datasets. Conclusions In summary, we show that adding intercellular dissimilarity can effectively improve accuracy and achieve robustness and that improved spectral clustering method outperforms the traditional spectral clustering method in grouping cells.

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