On regular graphs and coronas whose second largest eigenvalue does not exceed 1

2010 ◽  
Vol 58 (5) ◽  
pp. 545-554 ◽  
Author(s):  
Zoran Stanić
Keyword(s):  
Regular Graphs ◽  
Largest Eigenvalue ◽  
Second Largest Eigenvalue

scholarly journals NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

2010 ◽  
Vol 83 (1) ◽  
pp. 87-95
Author(s):  
KA HIN LEUNG ◽  
VINH NGUYEN ◽  
WASIN SO
Keyword(s):  
Regular Graph ◽  
Elementary Proof ◽  
Explicit Construction ◽  
Simple Graph ◽  
Regular Graphs ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Largest Eigenvalue ◽  
Image Position ◽  
Second Largest Eigenvalue

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

Eigenpolytopes of Distance Regular Graphs

1998 ◽  
Vol 50 (4) ◽  
pp. 739-755 ◽  
Author(s):  
C. D. Godsil
Keyword(s):  
Convex Hull ◽  
Orthogonal Projection ◽  
Adjacency Matrix ◽  
Gram Matrix ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Vertex Set ◽  
Distance Regular Graphs ◽  
Second Largest Eigenvalue

AbstractLet X be a graph with vertex set V and let A be its adjacency matrix. If E is the matrix representing orthogonal projection onto an eigenspace of A with dimension m, then E is positive semi-definite. Hence it is the Gram matrix of a set of |V| vectors in Rm. We call the convex hull of a such a set of vectors an eigenpolytope of X. The connection between the properties of this polytope and the graph is strongest when X is distance regular and, in this case, it is most natural to consider the eigenpolytope associated to the second largest eigenvalue of A. The main result of this paper is the characterisation of those distance regular graphs X for which the 1-skeleton of this eigenpolytope is isomorphic to X.

scholarly journals Regular graphs with small second largest eigenvalue

2013 ◽  
Vol 7 (2) ◽  
pp. 235-249 ◽  
Author(s):  
Tamara Koledin ◽  
Zoran Stanic
Keyword(s):  
Regular Graphs ◽  
Largest Eigenvalue ◽  
Second Largest Eigenvalue

We consider regular graphs with small second largest eigenvalue (denoted by ?2). In particular, we determine all triangle-free regular graphs with ?2 ? ?2, all bipartite regular graphs with ?2 ? ?3, and all bipartite regular graphs of degree 3 with ?2 ? 2.

scholarly journals Reflexive cacti: A survey

2016 ◽  
Vol 10 (2) ◽  
pp. 552-568
Author(s):  
Bojana Mihailovic ◽  
Marija Rasajski ◽  
Zoran Stanic
Keyword(s):  
Regular Graphs ◽  
Largest Eigenvalue ◽  
Second Largest Eigenvalue

A graph is called reexive if its second largest eigenvalue does not exceed 2. We survey the results on reexive cacti obtained in the last two decades. We also discuss various patterns of appearing of Smith graphs as subgraphs of reexive cacti. In the Appendix, we survey the recent results concerning reexive bipartite regular graphs.

scholarly journals Regular Factors of Regular Graphs from Eigenvalues

10.37236/431 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hongliang Lu
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Third ◽  
K Factor ◽  
Second Largest Eigenvalue

Let $r$ and $m$ be two integers such that $r\geq m$. Let $H$ be a graph with order $|H|$, size $e$ and maximum degree $r$ such that $2e\geq |H|r-m$. We find a best lower bound on spectral radius of graph $H$ in terms of $m$ and $r$. Let $G$ be a connected $r$-regular graph of order $|G|$ and $ k < r$ be an integer. Using the previous results, we find some best upper bounds (in terms of $r$ and $k$) on the third largest eigenvalue that is sufficient to guarantee that $G$ has a $k$-factor when $k|G|$ is even. Moreover, we find a best bound on the second largest eigenvalue that is sufficient to guarantee that $G$ is $k$-critical when $k|G|$ is odd. Our results extend the work of Cioabă, Gregory and Haemers [J. Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.

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.

On the estimation of the second largest eigenvalue of Markov ciphers

2016 ◽  
pp. n/a-n/a
Author(s):  
Weijia Xue ◽  
Tingting Lin ◽  
Xin Shun ◽  
Fenglei Xue ◽  
Xuejia Lai
Keyword(s):  
Largest Eigenvalue ◽  
Second Largest Eigenvalue
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