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In spectral graph theory, a Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders.

Examples of Ramanujan graphs include the clique, the biclique $K_{n,n}$, and the Petersen graph. As Murty's survey paper notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". As an example of this, a regular graph is Ramanujan if and only if its Ihara zeta function satisfies an analog of the Riemann hypothesis.

Definition

Let $G$ be a connected $d$-regular graph with $n$ vertices, and let ${\lambda }_0\ge {\lambda }_1\ge \dots \ge {\lambda }_{n-1}$ be the eigenvalues of the adjacency matrix of $G$. Because $G$ is connected and $d$-regular, its eigenvalues satisfy $d={\lambda }_0>{\lambda }_1$ $\ge \dots \ge {\lambda }_{n-1}\ge -d$. Whenever there exists ${\lambda }_i$ with $\left|{\lambda }_i\right|<d$, define

$\lambda (G)={\max }_{\left|{\lambda }_i\right|<d}\left|{\lambda }_i\right|.$

A $d$-regular graph $G$ is a Ramanujan graph if $\lambda (G)$ is defined and $\lambda (G)\le 2\sqrt{d-1}$.

A Ramanujan graph is characterized as a regular graph whose Ihara zeta function satisfies an analogue of the Riemann Hypothesis.

Extremality of Ramanujan graphs

For a fixed $d$ and large $n$, the $d$-regular, $n$-vertex Ramanujan graphs minimize $\lambda (G)$. If $G$ is a $d$-regular graph with diameter $m$, a theorem due to Nilli states

${\lambda }_1\ge 2\sqrt{d-1}-\frac{2\sqrt{d-1}-1}{\lfloor m/2\rfloor }.$

Whenever $G$ is $d$-regular and connected on at least three vertices, $\left|{\lambda }_1\right|<d$, and therefore $\lambda (G)\ge {\lambda }_1$. Let ${\mathcal{𝒢}}_n^d$ be the set of all connected $d$-regular graphs $G$ with at least $n$ vertices. Because the minimum diameter of graphs in ${\mathcal{𝒢}}_n^d$ approaches infinity for fixed $d$ and increasing $n$, Nilli's theorem implies an earlier theorem of Alon and Boppana which states

$\underset{n\to \mathrm{\infty }}{\lim }{\inf }_{G\in {\mathcal{𝒢}}_n^d}\lambda (G)\ge 2\sqrt{d-1}.$

Constructions

Constructions of Ramanujan graphs are often algebraic. Lubotzky, Phillips and Sarnak show how to construct an infinite family of p +1-regular Ramanujan graphs, whenever p ≡ 1 (mod 4) is a prime. Their proof uses the Ramanujan conjecture, which led to the name of Ramanujan graphs. Morgenstern extended the construction of Lubotzky, Phillips and Sarnak to all prime powers.

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

• Survey paper by M. Ram Murty