Forbidden pairs of disconnected graphs for 2‐factor of connected graphs

Graphs with many valencies and few eigenvalues

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2987 ◽

2015 ◽

Vol 28 ◽

Cited By ~ 1

Author(s):

Edwin Dam ◽

Jack Koolen ◽

Zheng-Jiang Xia

Keyword(s):

Connected Graphs ◽

Disconnected Graphs ◽

Switching Class ◽

Side Result

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph.

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Forbidden pairs of disconnected graphs for traceability in connected graphs

Discrete Mathematics ◽

10.1016/j.disc.2017.04.017 ◽

2017 ◽

Vol 340 (9) ◽

pp. 2194-2199

Author(s):

Junfeng Du ◽

Binlong Li ◽

Liming Xiong

Keyword(s):

Connected Graphs ◽

Disconnected Graphs

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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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