I am trying to search for an answer to the following question and I cannot find a straightforward answer.  What does the spectrum of the adjacency matrix (set of eigenvalues and their multiplicities) of a graph tell you about the structure of a graph?  Thank you!
Why do we care about eigenvalues of graphs?  Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important.  I always assumed that spectral graph theory extends graph theory by providing tools to prove things we couldn't otherwise, somewhat like how representation theory extends finite group theory.  But most results I see in spectral graph theory seem to concern eigenvalues  not as means to an end, but as objects of interest in their own right.  I also considered practical value as motivation, e.g. using a given set of eigenvalues to put bounds on essential properties of graphs, such as maximum vertex degree.  But I can't imagine a situation in which I would have access to a graph's eigenvalues before I would know much more elementary information like maximum vertex degree.  (EDIT: for example, dtldarek points out that $\lambda_2$ is related to diameter, but then why would we need $\lambda_2$ when we already have diameter?  Is this somehow conceptually beneficial?)     So, what is the meaning of graph spectra intuitively?  And for what practical purposes are they used?  Why is finding the eigenvalues of a graph's adjacency/Laplacian matrices more than just a novel problem?
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I updated some entries in the  at C:\WINDOWS\System32\drivers\etc.  As a test, I decided to make Google point to 127.0.0.1.  # Additionally, comments (such as these) may be inserted on individual # lines or following the machine name denoted by a '#' symbol. # # For example: # #      102.54.94.97     rhino.acme.com          # source server #       38.25.63.10     x.acme.com              # x client host  127.0.0.1       localhost ::1             localhost 127.0.0.1       google.com   This works fine on Windows Vista, but not on Widows 7. When I type google.com into my browser, it goes directly to Google's website.  I am not using a proxy server, and I think there might be some temporary DNS settings that need to be flushed, but I'm not sure.  Does anyone know how to fix this? Order of elements in $Z_n$ I have this question:     Let $x, n$ be integers with $n \geq 2$ and $n$ not dividing $x$.  Show that the order o($\bar{x}$) of $x \in Z_n$ is   $o(\bar{x})= \frac{n}{HCF(x, n)}$   I've been thinking about it for ages but I still don't get why. A hint would be appreciated. Are there any age limits (formal, informal, or guidelines) that schools use when deciding to accept someone into a PhD program? I'm most curious about the upper age limits. For example, will most schools accept someone in their 40's? How about their 50's? I'm trying to write a blog post which includes a code segment inside a &lt;pre&gt; tag. The code segment includes a generic type and uses &lt;&gt; to define that type. This is what the segment looks like:  &lt;pre&gt;     PrimeCalc calc = new PrimeCalc();     Func&lt;int, int&gt; del = calc.GetNextPrime; &lt;/pre&gt;   The resulting HTML removes the &lt;&gt; and ends up like this:  PrimeCalc calc = new PrimeCalc(); Func del = calc.GetNextPrime;   How do I escape the &lt;&gt; so they show up in the HTML? Dynamic Paint not showing in rendered image In the sentence: "Drinking and driving IS dangerous", why does the verb "to be" need to be singular? In the sentence: "Drinking and driving is dangerous". Drinking and driving are both subjects and the word dangerous is the object (I think) but why do I have to use the verb to be in singular? Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ Integral inequality with a strange condition Let $f$ be a continuously differentiable real valued function on $[0,1]$.  It is given that $\displaystyle \int_{\frac{1}{3}}^{\frac{2}{3}}f(x) dx=0$     Find the minimum value of $\dfrac{\int_{0}^{1} (f'(x))^2 dx}{\left( \int_{0}^{1} f(x) dx \right)^2}$   I tried to use Cauchy-Schwarz to show that  $$\frac{\int_{0}^{1} (f'(x))^2 dx}{\left( \int_0^1 f(x) dx \right)^2} \ge \frac{\left( \int_0^1 \bigl| f(x)f'(x) \bigr| dx \right)^2}{ \left( \int_0^1 f(x) dx \right)^2} \ge \frac{ f(1)^2 - f(0)^2}{2 \left( \int_0^1 f^2 (x) dx \right)^2}$$ But I can't proceed from here.  Also, I don't know how to use the condition  $\int_{1/3}^{2/3}f(x) dx=0$ What does the $\ni$ (backwards element of) symbol mean?  It doesn't appear in the , and a Google search for "backwards element of" or "backwards epsilon" turns up contradictory (or unreliable) information.  It seems it can mean both "such that", or "contains as an element".  Is this correct, and if so, which is the more common usage? The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ Best way to remove an event handler in jQuery? Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.  $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$  Perhaps someone can help me out I don't understand how to move forward from here: $$\sum\limits_{k=1}^{n+1}{\frac{1}{\sqrt{k}}+\frac{1}{\sqrt{n+1}}\ge \sqrt{n+1}}$$ proof and explanation would greatly be appreciated :) Thanks :)  EDIT sorry meant GE not = fixed :) Can anyone tell me if I should use inspire or inspires in this phrase?     An extraordinary leader whose vision, values, integrity and boundless curiosity inspires all who follow in his footsteps. Clothing store access in Pokémon X & Y Once the player obtains access to Lumiose City for the first time, they'll be able to visit Boutique Couture from the city's Vernal Avenue (South fork, connecting from South Boulevard). Upon entering the store without performing any other actions, the player will be turned away and informed they "aren't stylish enough" and that they should visit more places in Lumiose City before visiting again.  This description is dubious, and doesn't inform the player of what they need to accomplish to gain access. Specifically:  Even beating the game (completing the Pokémon League) fails to allow access, as does modifying the player's look, visiting most shops in all areas of Lumiose City, and performing all hair stylist, PR studio, and associated actions. Completing a set of triple battles in Restaurant Le Wow did set the event flag that allowed access in my copy of Y, but it is not clear if this is the only requirement for access.  So other players may benefit, and so I know how to repeat this process if I obtain a copy of X or the future release of Z, what exact requirements yield access to this store for the first time? How can I copy files from A to B, only including files that exist in B? I'm on Windows 7 Professional, and this is my scenario :  Folder "Asd" (C:\)              Folder "Asd" (D:\) File 1                          File 1  File 2                          File 3  File 3                          File 4 File 4 File 5   I'd like to select the folder on C:\ (without selecting each single file, of course) and copy it over the same folder on D:. Usually, it will copy all files. But what I'm looking for is to copy only files (from C:) that are contained on folder d:.  So in my case I'd like to replace only File 1, File 3 and File 4.  File 2 and File 5 must be ignored.  Is it possible? How to configure Micromax 352g USB Modem? I want the driver software for my Micromax 352g USB modem for Ubuntu 12.04 LTS. 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