scholarly journals Eigenvalue bounds for the signless laplacian

2007 ◽  
pp. 11-27 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Peter Rowlinson ◽  
Slobodan Simic
Keyword(s):  
Previous Survey ◽  
Eigenvalue Bounds ◽  
Largest Eigenvalue ◽  
Signless Laplacian

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.

Graphs with the second signless Laplacian eigenvalue ≤ 4

2021 ◽  
Vol 10 (1) ◽  
pp. 131-152
Author(s):  
Stephen Drury
Keyword(s):  
General Solution ◽  
Knapsack Problem ◽  
Laplacian Eigenvalue ◽  
Simple Graphs ◽  
Largest Eigenvalue ◽  
Signless Laplacian ◽  
Second Largest Eigenvalue

Abstract We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

scholarly journals Edge bipartiteness and signless Laplacian spread of graphs

2012 ◽  
Vol 6 (1) ◽  
pp. 31-45 ◽  
Author(s):  
Yi-Zheng Fan ◽  
Shaun Fallat
Keyword(s):  
Connected Graph ◽  
Eigenvalue Bounds ◽  
Signless Laplacian

Let G be a connected graph, and let ?b(G) and SQ(G) be the edge bipartiteness and the signless Laplacian spread of G, respectively. We establish some important relationships between ?b(G) and SQ(G), and prove SQ(G)?2 (1+cos?/n) with equality if and only if G = Pn or G = Cn in case of odd n. In addition, we show that if G?Pn or G?C2k+1; then SQ(G)?4; with equality if and only if G is one of the following graphs: K1,3, K4, two triangles connected by an edge, and Cn for even n. As a consequence, we prove a conjecture of Cvetkovic, Rowlinson and Simic on minimal signless Laplacian spread [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd), 81(95)(2007), 11-27].

scholarly journals On three conjectures involving the signless Laplacian spectral radius of graphs

2009 ◽  
Vol 85 (99) ◽  
pp. 35-38 ◽  
Author(s):  
Lihua Feng ◽  
Guihai Yu
Keyword(s):  
Spectral Radius ◽  
Eigenvalue Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

We study the signless Laplacian spectral radius of graphs and prove three conjectures of Cvetkovic, Rowlinson, and Simic [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math., Nouv. S?r. 81(95) (2007), 11-27].

scholarly journals On the second largest eigenvalue of the signless Laplacian

2013 ◽  
Vol 438 (3) ◽  
pp. 1215-1222 ◽  
Author(s):  
Leonardo Silva de Lima ◽  
Vladimir Nikiforov
Keyword(s):  
Largest Eigenvalue ◽  
Signless Laplacian ◽  
Second Largest Eigenvalue

scholarly journals Eigenvalue Bounds for the Signless $p$-Laplacian

10.37236/6683 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Elizandro Max Borba ◽  
Uwe Schwerdtfeger
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Unit Sphere ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Eigenvalue Bounds ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Largest Eigenvalues

We consider the signless $p$-Laplacian $Q_p$ of a graph, a generalisation of the quadratic form of the signless Laplacian matrix (the case $p=2$). In analogy to Rayleigh's principle the minimum and maximum of $Q_p$ on the $p$-norm unit sphere are called its smallest and largest eigenvalues, respectively. We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a graph parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, at $p=1$ upper and lower bounds coincide.

scholarly journals Maxima of the signless Laplacian spectral radius for planar graphs

2015 ◽  
Vol 30 ◽  
pp. 795-811 ◽  
Author(s):  
Guanglong Yu ◽  
Jianyong Wong ◽  
Shu-guang Guo
Keyword(s):  
Spectral Radius ◽  
Planar Graphs ◽  
Laplacian Spectral Radius ◽  
Largest Eigenvalue ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
The Largest Eigenvalue

The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph $K_{2}\nabla P_{n-2}$ has the maximal signless Laplacian spectral radius among all planar graphs of order $n\geq 456$.

Ordering cacti with signless Laplacian spread

2018 ◽  
Vol 34 ◽  
pp. 609-619 ◽  
Author(s):  
Zhen Lin ◽  
Shu-Guang Guo
Keyword(s):  
Connected Graph ◽  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The Difference ◽  
The Largest Eigenvalue

A cactus is a connected graph in which any two cycles have at most one vertex in common. The signless Laplacian spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated signless Laplacian matrix. In this paper, all cacti of order n with signless Laplacian spread greater than or equal to n − 1/2 are determined.

scholarly journals An asymptotically tight bound on the q-index of graphs with forbidden cycles

2014 ◽  
Vol 95 (109) ◽  
pp. 189-199 ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Asymptotic Solution ◽  
Tight Bound ◽  
Largest Eigenvalue ◽  
Signless Laplacian ◽  
Q Index ◽  
The Largest Eigenvalue

Let G be a graph of order n and let q(G) be the largest eigenvalue of the signless Laplacian of G. It is shown that if k ? 2, n > 5k2, and q(G) ? n + 2k ? 2, then G contains a cycle of length l for each l ? {3, 4,..., 2k + 2}. This bound on q(G) is asymptotically tight, as the graph Kk ?Kn?k contains no cycles longer than 2k and q(Kk ?Kn?k) > n + 2k?2?2k(k ? 1)/ n+2k?3. The main result gives an asymptotic solution to a recent conjecture about the maximum q(G) of a graph G with forbidden cycles. The proof of the main result and the tools used therein could serve as a guidance to the proof of the full conjecture.

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