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.

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 Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals A note on a walk-based inequality for the index of a signed graph

2021 ◽  
Vol 9 (1) ◽  
pp. 19-21
Author(s):  
Zoran Stanić
Keyword(s):  
Adjacency Matrix ◽  
Signed Graph ◽  
Arbitrary Length ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

Abstract We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.

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 The Laplacian Spread of Tricyclic Graphs

10.37236/169 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yanqing Chen ◽  
Ligong Wang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Fixed Order ◽  
Tricyclic Graphs ◽  
The Difference ◽  
The Largest Eigenvalue

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

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.

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