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.

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.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

On the Non-Existence of a Type of Regular Graphs of Girth 5

1967 ◽  
Vol 19 ◽  
pp. 644-648 ◽  
Author(s):  
William G. Brown
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Minimal Graph ◽  
Image Position

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that1.1Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

ON CONSTRUCTION OF ALMOST-RAMANUJAN GRAPHS

2009 ◽  
Vol 01 (02) ◽  
pp. 193-203
Author(s):  
HE SUN ◽  
HONG ZHU
Keyword(s):  
Explicit Construction ◽  
Similar Technique ◽  
Largest Eigenvalue ◽  
Ramanujan Graphs ◽  
Big Graph ◽  
Second Largest Eigenvalue

O. Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of d-regular expanders with the second largest eigenvalue O(d-1/3). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d-1/2). Such graphs are called Ramanujan graphs. Recently, zig-zag product has been generalized by A. Ben-Aroya and A. Ta-Shma. Using this technique, they present a family of expanders with the second largest eigenvalue d-1/2 + o(1), what they call almost-Ramanujan graphs. However, their construction relies on local invertible functions and the dependence between the big graph and several small graphs, which makes the construction more complicated. In this paper, we shall give a generalized theorem of zig-zag product. Specifically, the zig-zag product of one "big" graph and several "small" graphs with the same size will be formalized. By choosing the big graph and several small graphs individually, we shall present a family of fully explicitly almost-Ramanujan graphs with locally invertible function waived.

scholarly journals Integral Circulant Ramanujan Graphs of Prime Power Order

10.37236/3159 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
T. A. Le ◽  
J. W. Sander
Keyword(s):  
Regular Graph ◽  
Prime Power ◽  
Cayley Graphs ◽  
Complete List ◽  
Prime Power Order ◽  
Circulant Graphs ◽  
Largest Eigenvalue ◽  
Ramanujan Graphs ◽  
Vertex Set

A connected $\rho$-regular graph $G$ has largest eigenvalue $\rho$ in modulus. $G$ is called Ramanujan if it has at least $3$ vertices and the second largest modulus of its eigenvalues is at most $2\sqrt{\rho-1}$. In 2010 Droll classified all Ramanujan unitary Cayley graphs. These graphs of type ${\rm ICG}(n,\{1\})$ form a subset of the class of integral circulant graphs ${\rm ICG}(n,{\cal D})$, which can be characterised by their order $n$ and a set $\cal D$ of positive divisors of $n$ in such a way that they have vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{(a,b):\, a,b\in\mathbb{Z}/n\mathbb{Z} ,\, \gcd(a-b,n)\in {\cal D}\}$. We extend Droll's result by drawing up a complete list of all graphs ${\rm ICG}(p^s,{\cal D})$ having the Ramanujan property for each prime power $p^s$ and arbitrary divisor set ${\cal D}$.  

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 Unicyclic reflexive graphs with seven loaded vertices of the cycle

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 257-268
Author(s):  
Tamara Koledin ◽  
Zoran Radosavljevic
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Largest Eigenvalue ◽  
Reflexive Graph ◽  
Second Largest Eigenvalue

A simple graph is reflexive if the second largest eigenvalue of its (0, 1)- adjacency matrix does not exceed 2. A vertex of the cycle of unicyclic simple graph is said to be loaded if its degree is greater than 2. In this paper we establish that the length of the cycle of unicyclic reflexive graph with seven loaded vertices is at most 10 and find all such graphs with the length of the cycle 10, 9 and 8.

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