Eigenvalues and expansion of bipartite graphs

Høholdt, Tom; Janwa, Heeralal

doi:10.1007/s10623-011-9598-6

Cited by 13 publications

(13 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we studied b(k, θ), the maximum number of vertices in bipartite regular graph of valency k whose second largest eigenvalue is at most θ. Our results extend previous work from [9,18,19,24,29]. Our general bound for b(k, θ) is attained whenever there exists a bipartite distance-regular graph of valency k, second largest eigenvalue θ, girth g and diameter d with g ≥ 2d − 2.…”

Section: Discussionsupporting

confidence: 83%

A spectral version of the Moore problem for bipartite regular graphs

Cioabă¹,

Koolen²,

Nozaki³

2019

Algebraic Combinatorics

View full text Add to dashboard Cite

Let b(k, θ) be the maximum order of a connected bipartite k-regular graph whose second largest eigenvalue is at most θ. In this paper, we obtain a general upper bound for b(k, θ) for any 0 ≤ θ < 2 √ k − 1. Our bound gives the exact value of b(k, θ) whenever there exists a bipartite distance-regular graph of degree k, second largest eigenvalue θ, diameter d and girth g such that g ≥ 2d − 2. For certain values of d, there are infinitely many such graphs of various valencies k. However, for d = 11 or d ≥ 15, we prove that there are no bipartite distance-regular graphs with g ≥ 2d − 2.

show abstract

Section: Discussionsupporting

confidence: 83%

A spectral version of the Moore problem for bipartite regular graphs

Cioabă¹,

Koolen²,

Nozaki³

2019

Algebraic Combinatorics

View full text Add to dashboard Cite

show abstract

“…For evolution of networks (see for example Sharan et al, 2005; Mazurie et al, 2010). For bipartite networks (Janwa and Lal, 2003; Høholdt and Janwa, 2012). For Spectral methods (Cvetković et al, 1980; Lubotzky et al, 1988; Lubotzky, 1994, 2012; Chung, 1997; Davidoff et al, 2003; Sarnak, 2004; Chung and Lu, 2006; Spielman and Teng, 2011; Janwa and Rangachari, 2015).…”

Section: Bibliographic Notesmentioning

confidence: 99%

On the Origin of Biomolecular Networks

et al. 2019

Self Cite

View full text Add to dashboard Cite

Biomolecular networks have already found great utility in characterizing complex biological systems arising from pairwise interactions amongst biomolecules. Here, we explore the important and hitherto neglected role of information asymmetry in the genesis and evolution of such pairwise biomolecular interactions. Information asymmetry between sender and receiver genes is identified as a key feature distinguishing early biochemical reactions from abiotic chemistry, and a driver of network topology as biomolecular systems become more complex. In this context, we review how graph theoretical approaches can be applied not only for a better understanding of various proximate (mechanistic) relations, but also, ultimate (evolutionary) structures encoded in such networks from among all types of variations they induce. Among many possible variations, we emphasize particularly the essential role of gene duplication in terms of signaling game theory , whereby sender and receiver gene players accrue benefit from gene duplication, leading to a preferential attachment mode of network growth. The study of the resulting dynamics suggests many mathematical/computational problems, the majority of which are intractable yet yield to efficient approximation algorithms, when studied through an algebraic graph theoretic lens. We relegate for future work the role of other possible generalizations, additionally involving horizontal gene transfer, sexual recombination, endo-symbiosis, etc., which enrich the underlying graph theory even further.

show abstract

“…The lower bound in Theorem 1.1 will be proved by bounding |N (S)| from below for an arbitrary S ⊆ V with 1 ≤ |S| ≤ |V | 2 . Let X = S ∩ P and Y = S ∩ B, and let x, y, x ′ , and y ′ be defined as in (14). Then |S| = x + y and |N (S)| = x ′ + y ′ .…”

Section: Lower Boundmentioning

confidence: 99%

“…Let S be an arbitrary subset of V with 1 ≤ |S| ≤ |V | 2 , and let X = S ∩ P and Y = S ∩ B. Let x, y, x ′ , and y ′ be defined as in (14). In view of (1), the parameters of U are (v, b, r, k, λ) = (n 3 + 1, n 4 − n 3 + n 2 , n 4 − n 3 , n 3 − n, n 4 − n 3 − n 2 + 1).…”

Section: Proof Of Theorem 16mentioning

confidence: 99%

The vertex-isoperimetric number of the incidence and non-incidence graphs of unitals

Hui

Surani

Zhou

2018

Des. Codes Cryptogr.

View full text Add to dashboard Cite

We derive upper and lower bounds for the vertex-isoperimetric number of the incidence graphs of unitals and determine its order of magnitude. In the case when a unital contains sufficiently large arcs, these bounds agree and give rise to the precise value of this parameter. In particular, we obtain the exact value of the vertex-isoperimetric number of the incidence graphs of classical unitals and a certain subfamily of BM-unitals. In the case when the maximum size of arcs in the unital is relatively small, we obtain an upper bound for this parameter in terms of the vertex-isoperimetric number of the incidence graph. We also determine the exact value of the vertex-isoperimetric number of the non-incidence graph of any unital.

show abstract

Eigenvalues and expansion of bipartite graphs

Cited by 13 publications

References 25 publications

A spectral version of the Moore problem for bipartite regular graphs

A spectral version of the Moore problem for bipartite regular graphs

On the Origin of Biomolecular Networks

The vertex-isoperimetric number of the incidence and non-incidence graphs of unitals

Contact Info

Product

Resources

About