Further results on the least eigenvalue of connected graphs

Miroslav Petrović; Tatjana Aleksić; Slobodan Simić

doi:10.1016/j.laa.2011.04.030

Least eigenvalue of the connected graphs whose complements are cacti

Open Mathematics ◽

10.1515/math-2019-0097 ◽

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.

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Controllable graphs with least eigenvalue at least -2

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm110909022c ◽

2011 ◽

Vol 5 (2) ◽

pp. 165-175 ◽

Cited By ~ 12

Author(s):

Dragos Cvetkovic ◽

Peter Rowlinson ◽

Zoran Stanic ◽

Myung-Gon Yoon

Keyword(s):

Control Theory ◽

Connected Graphs ◽

Largest Eigenvalue ◽

Least Eigenvalue ◽

Second Largest Eigenvalue

Connected graphs whose eigenvalues are distinct and main are called controllable graphs in view of certain applications in control theory. We give some general characterizations of the controllable graphs whose least eigenvalue is bounded from below by - 2; in particular, we determine all the controllable exceptional graphs. We also investigate the controllable graphs whose second largest eigenvalue does not exceed 1.

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

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm130710014z ◽

2013 ◽

Vol 7 (2) ◽

pp. 250-261 ◽

Cited By ~ 5

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).

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Minimal least eigenvalue of connected graphs of order n and size m = n + k (5 ⩽ k ⩽ 8)

Frontiers of Mathematics in China ◽

10.1007/s11464-019-0805-5 ◽

2019 ◽

Vol 14 (6) ◽

pp. 1213-1230

Author(s):

Xin Li ◽

Jiming Guo ◽

Zhiwen Wang

Keyword(s):

Connected Graphs ◽

Least Eigenvalue

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The Formula to Count The Number of Vertices Labeled Order Six Connected Graphs with Maximum Thirty Edges without Loops

Journal of Physics Conference Series ◽

10.1088/1742-6596/1751/1/012023 ◽

2021 ◽

Vol 1751 ◽

pp. 012023

Author(s):

F C Puri ◽

Wamiliana ◽

M Usman ◽

Amanto ◽

M Ansori ◽

...

Keyword(s):

Connected Graphs

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On minimum algebraic connectivity of graphs whose complements are bicyclic

Open Mathematics ◽

10.1515/math-2019-0119 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1490-1502 ◽

Cited By ~ 1

Author(s):

Jia-Bao Liu ◽

Muhammad Javaid ◽

Mohsin Raza ◽

Naeem Saleem

Keyword(s):

Connected Graph ◽

Diffusion Processes ◽

Laplacian Matrix ◽

Group Differences ◽

Algebraic Connectivity ◽

Connected Graphs ◽

Bicyclic Graph ◽

Smallest Eigenvalue ◽

Unique Graph ◽

The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

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