scholarly journals The Least Eigenvalue of the Graphs Whose Complements Are Connected and Have Pendent Paths

2018 ◽  
Vol 8 (4) ◽  
pp. 303-308 ◽  
Author(s):  
Chen Wang ◽  
Guidong Yu ◽  
Wei Sun ◽  
Jinde Cao
Keyword(s):  
Adjacency Matrix ◽  
Minimum Eigenvalue ◽  
Least Eigenvalue ◽  
Unique Graph ◽  
Adjacent Relation

Abstract The adjacency matrix of a graph is a matrix which represents adjacent relation between the vertices of the graph. Its minimum eigenvalue is defined as the least eigenvalue of the graph. Let Gn be the set of the graphs of order n, whose complements are connected and have pendent paths. This paper investigates the least eigenvalue of the graphs and characterizes the unique graph which has the minimum least eigenvalue in Gn.

scholarly journals The Optimal Graph Whose Least Eigenvalue is Minimal among All Graphs via 1-2 Adjacency Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Lubna Gul ◽  
Gohar Ali ◽  
Usama Waheed ◽  
Nudrat Aamir
Keyword(s):  
Adjacency Matrix ◽  
Least Eigenvalue ◽  
New Type ◽  
Unique Graph ◽  
Optimal Graph

All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 ,   i f   v i = v j   o r     d v i , v j ≥ 2 1 ,   i f     d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 ,   i f   v i = v j   o r     d v i , v j ≥ 3 1 ,   i f     d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .

scholarly journals Least eigenvalue of the connected graphs whose complements are cacti

2019 ◽  
Vol 17 (1) ◽  
pp. 1319-1331
Author(s):  
Haiying Wang ◽  
Muhammad Javaid ◽  
Sana Akram ◽  
Muhammad Jamal ◽  
Shaohui Wang
Keyword(s):  
Linear Algebra ◽  
Adjacency Matrix ◽  
Connected Graphs ◽  
Least Eigenvalue ◽  
The Family

Abstract Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each Cc ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, where $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.

scholarly journals On the nullity of connected graphs with least eigenvalue at least -2

2013 ◽  
Vol 7 (2) ◽  
pp. 250-261 ◽  
Author(s):  
Jiang Zhou ◽  
Lizhu Sun ◽  
Hongmei Yao ◽  
Changjiang Bu
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Line Graphs ◽  
Connected Graphs ◽  
Least Eigenvalue

Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by ?(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ? L, if ?(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph).

scholarly journals Line graphs of complex unit gain graphs with least eigenvalue -2

2021 ◽  
Vol 37 (37) ◽  
pp. 14-30
Author(s):  
Maurizio Brunetti ◽  
Francesco Belardo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Multiplicative Group ◽  
Line Graph ◽  
Line Graphs ◽  
Signed Graphs ◽  
The Real ◽  
Least Eigenvalue ◽  
Gain Graphs ◽  
Gain Graph

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

On the Estrada index of cacti

2014 ◽  
Vol 27 ◽  
Author(s):  
Erfang Shan ◽  
Hongzhuan Wang ◽  
Liying Kang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Common Vertex ◽  
Estrada Index ◽  
Unique Graph ◽  
Simple Connected Graph

Let G be a simple connected graph on n vertices and λ_1, λ_2, . . . , λ_n be the eigenvalues of the adjacency matrix of G. The Estrada index of G is defined as EE(G) = \sum_{i=1}^n e^{λi}. A cactus is a connected graph in which any two cycles have at most one common vertex. In this work, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cycles was determined. Also, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cut edges was determined.

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

2019 ◽  
Vol 10 (3) ◽  
pp. 565-573
Author(s):  
Keerthi G. Mirajkar ◽  
Bhagyashri R. Doddamani
Keyword(s):  
Adjacency Matrix ◽  
Eigen Values

Energy of spherical fuzzy graphs

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.

Minimum Eigenvalue Separation

1992 ◽  
Author(s):  
Beresford Parlett ◽  
Tzon-Tzer Lu
Keyword(s):  
Minimum Eigenvalue

Mathematics of networks

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.

scholarly journals A Note on the Estrada Index of the Aα-Matrix

Mathematics ◽  
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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