Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic

Javaid, Muhammad

doi:10.3390/math5010018

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…Equation ( 4) is called the eigen equation for the graph G. Now, for any arbitrary unit vector X ∈ R n , λ min (G) ≤ X T B(G)X, (5) and it will be equal iff X is a least eigenvector of G. Now, let G c denote the complement of a graph G. One can easily prove that B(G C ) � J − I − B(G), where I and J denote the identity matrix and the ones matrix of the same size as B(G). Now, for any vector X,…”

Section: Resultsmentioning

confidence: 99%

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The Least Eigenvalue of the Complement of the Square Power Graph of G

et al. 2021

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Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .

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Section: Resultsmentioning

confidence: 99%

“…Hong and Shu [3] discussed the sharp lower bounds of the least eigenvalue of planar graphs. Javid [4,5] discussed the minimizing graphs of the connected graphs whose complements are bicyclic (with two cycles) and explained the characterization of the minimizing graph of the connected graphs whose complement is bicyclic.…”

Section: Introductionmentioning

confidence: 99%

The Least Eigenvalue of the Complement of the Square Power Graph of G

et al. 2021

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“…ere are many results in the literature concerning the largest eigenvalue (spectral radius) of simple graph; see, e.g., [2] or [1]. Javaid examined different families of graphs to pick optimal graph with respect to least eigenvalues via usual adjacency matrix in their respective graph classes in [3][4][5]. Lubna et al examined square power graph of G for their least eigenvalue [6].…”

Section: Introductionmentioning

confidence: 99%

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

Gul

Ali

Waheed

et al. 2021

Journal of Mathematics

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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 .

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“…After it, the question is raised to investigate the minimizing graphs in the collection of connected graphs such that the complement of each graph contains the cliques of small sizes. Motivated by it, the minimizing graph in the collection of connected graphs such that the complement of each graph is trees, unicyclic or bicyclic are characterized by Fan, Zhang, Wang, Li and Javaid, see [15][16][17][18]. For further study, we refer to [19 -22].…”

Section: Introductionmentioning

confidence: 99%

Least eigenvalue of the connected graphs whose complements are cacti

et al. 2019

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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.

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Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic

Cited by 5 publications

References 16 publications

The Least Eigenvalue of the Complement of the Square Power Graph of G

The Least Eigenvalue of the Complement of the Square Power Graph of G

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

Least eigenvalue of the connected graphs whose complements are cacti

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