On the multiplicity of Laplacian eigenvalues for unicyclic graphs

2022 ◽  
pp. 1-20
Author(s):  
Fei Wen ◽  
Qiongxiang Huang
Keyword(s):  
Unicyclic Graphs ◽  
Laplacian Eigenvalues

scholarly journals The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs

2019 ◽  
Vol 35 ◽  
pp. 449-467
Author(s):  
Zhibin Du
Keyword(s):  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian ◽  
Pendent Vertices ◽  
Eigenvalues Of Graphs

Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.

scholarly journals On Graphs with Three or Four Distinct Normalized Laplacian Eigenvalues

2019 ◽  
Vol 26 (01) ◽  
pp. 65-82 ◽  
Author(s):  
Xueyi Huang ◽  
Qiongxiang Huang
Keyword(s):  
Bipartite Graphs ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Laplacian Eigenvalues

We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.

scholarly journals Some results on the Laplacian eigenvalues of unicyclic graphs

2009 ◽  
Vol 430 (8-9) ◽  
pp. 2080-2093 ◽  
Author(s):  
Jianxi Li ◽  
Wai Chee Shiu ◽  
Wai Hong Chan
Keyword(s):  
Unicyclic Graphs ◽  
Laplacian Eigenvalues

On the Laplacian Eigenvalues of Gn,p

2007 ◽  
Vol 16 (6) ◽  
pp. 923-946 ◽  
Author(s):  
AMIN COJA-OGHLAN
Keyword(s):  
Random Graphs ◽  
Spectral Gap ◽  
Degree Sequences ◽  
Laplacian Eigenvalues ◽  
Combinatorial Laplacian ◽  
Laplacian Spectra

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian $\LL(\gnp)$ is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of $\LL(\core(G))$ is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of $\LL(\gnp)$ is 1-O (d−1/2) w.h.p.

On the Reverse Wiener Indices of Unicyclic Graphs

2008 ◽  
Vol 106 (2) ◽  
pp. 293-306 ◽  
Author(s):  
Zhibin Du ◽  
Bo Zhou
Keyword(s):  
Unicyclic Graphs

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

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