On the structure of the adjacency matrix of the line digraph of a regular digraph

Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of unsymmetric digraphs.

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TRANSMISSION FAULT-TOLERANCE OF ITERATED LINE DIGRAPHS

Journal of Interconnection Networks ◽

10.1142/s021926590400126x ◽

2004 ◽

Vol 05 (04) ◽

pp. 475-487

Author(s):

LU RUAN ◽

SHITOU HAN ◽

DEYING LI ◽

HUNG Q. NGO ◽

SCOTT C.-H. HUANG

Keyword(s):

Fault Tolerance ◽

Interconnection Network ◽

Line Digraph ◽

Kautz Digraphs ◽

Network Topologies ◽

De Bruijn ◽

De Bruijn Digraphs ◽

Regular Digraph

The main result of this paper states that, if every cyclic modification of a d-regular digraph has super line-connectivity d, then the line digraph also has super line-connectivity d. Since many well-known interconnection network topologies, such as the Kautz digraphs, the de Bruijn digraphs, etc., can be constructed by iterating the line digraph construction, our result leads to several known and new connectivity results for these topologies, as shown later in the paper.

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Normally Regular Digraphs

The Electronic Journal of Combinatorics ◽

10.37236/4798 ◽

2015 ◽

Vol 22 (4) ◽

Cited By ~ 1

Author(s):

Leif K Jørgensen

Keyword(s):

Directed Graph ◽

Adjacency Matrix ◽

Existence Results ◽

Association Schemes ◽

Strongly Regular Graphs ◽

Regular Graphs ◽

Combinatorial Objects ◽

Strongly Regular ◽

Regular Digraph ◽

Structural Characterizations

A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $\mu$ common out-neighbours, a pair of vertices connected by an edge in one direction have $\lambda$ common out-neighbours and a pair of vertices connected by edges in both directions have $2\lambda-\mu$ common out-neighbours. We often assume that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regular digraph is normal. A connected $k$-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than $k$ are on one circle in the complex plane. We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs are Cayley graphs of abelian groups and the construction is then based on a generalization of difference sets.We also show connections to other combinatorial objects: strongly regular graphs, symmetric 2-designs and association schemes.

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Bounds for the Eigen Values and Energy of Degree Product Adjacency Matrix of A Graph

Journal of Computer and Mathematical Sciences ◽

10.29055/jcms/1038 ◽

2019 ◽

Vol 10 (3) ◽

pp. 565-573

Author(s):

Keerthi G. Mirajkar ◽

Bhagyashri R. Doddamani

Keyword(s):

Adjacency Matrix ◽

Eigen Values

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Energy of spherical fuzzy graphs

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.26 ◽

2020 ◽

pp. 321-332

Author(s):

S. Yahya Mohamed ◽

A. Mohamed Ali

Keyword(s):

Adjacency Matrix ◽

Upper Bounds ◽

Fuzzy Graph ◽

Lower And Upper Bounds ◽

Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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Mathematics of networks

10.1093/oso/9780198805090.003.0006 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Random Walks ◽

Adjacency Matrix ◽

Graph Partitioning ◽

Dynamic Networks ◽

Graph Laplacian ◽

Network Visualization ◽

Final Part ◽

Bipartite Networks ◽

Component Structure ◽

Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

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A Note on the Estrada Index of the Aα-Matrix

Mathematics ◽

10.3390/math9080811 ◽

2021 ◽

Vol 9 (8) ◽

pp. 811

Author(s):

Jonnathan Rodríguez ◽

Hans Nina

Keyword(s):

Lower Bounds ◽

Adjacency Matrix ◽

Estrada Index

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A(G) and D(G) and defined the Aα-matrix for every real α∈[0,1] as: Aα(G)=αD(G)+(1−α)A(G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the Aα-matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established.

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Properties of adjacency matrix of the directed cyclic friendship graph

10.1063/5.0042158 ◽

2021 ◽

Author(s):

Nanda Anzana ◽

Siti Aminah ◽

Suarsih Utama

Keyword(s):

Adjacency Matrix

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Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽

10.3390/e23080976 ◽

2021 ◽

Vol 23 (8) ◽

pp. 976

Author(s):

R. Aguilar-Sánchez ◽

J. Méndez-Bermúdez ◽

José Rodríguez ◽

José Sigarreta

Keyword(s):

Random Matrix ◽

Adjacency Matrix ◽

Matrix Theory ◽

Computational Study ◽

Random Networks ◽

Geometric Graphs ◽

Theory Approach ◽

Complexity Measures ◽

Random Geometric Graphs ◽

Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

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