If you have a graph with 1 node and no edges and you then remove this 1 node, is the remaining a graph with 0 nodes or is it not anymore a graph?  Alternative formulation: Is there something such as a empty null graph?
Standard definition of graph says that it is an ordered pair G=(N,L) where N is set of nodes and L is set of lines which connect the nodes.   From what I've read, the set L can be empty, but can set N be empty too?  I'm mainly asking this because for my exam, I found numerous problems that go like this: Graph X is given. How many subgraphs with property Y exist? Often the number depends on the definition of graph.
My book gives the following definition:     Let $(M,d)$ be a metric space, and let $\mathcal{T}$ be the collection of all subsets of $M$ that are open in the metric space sense...  $\mathcal{T}$ is called the topology generated by d   Then it later says to prove things about two different metrics generating the same topology. However I'm very confused about what all this means.   When I read "all subsets," I think of the theorem that says a set with $n$ elements will have $2^n$ subsets. (ex. all the subsets of $\{1,2,3\}$ are $\{\emptyset, \{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\} \}$). So I think "all subsets that are open" would be like, If M is the set (1,5), then an open subset would be any set $(a,b)$ where $1&lt;a&lt;b&lt;5$, and then all subsets would be all possible union combinations of these "$(a,b)$" sets.     Its seems to me that metrics have nothing to do with subsets. Clearly there is something i'm not understanding. Please Help me.   It would be very enlightening, if someone could give me some simple examples of 1 set and 2 metrics generating two different typologies. If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $ I have come across this .  Has anyone tried anything similar or does it work for you? Can you tell if an email customer / user enters is correct &amp; exists? What are all the marriages and child relationships following the final chapter? In the last chapter of Naruto, some characters got married, and some of them had kids. What is the entire husband-wife-kids set? I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. Check whether $\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{1}{\left(m+n\right)^2}$ converges or NOT? Compute $\int_{0}^{\infty}\frac{x\sin 2x}{9+x^{2}} \, dx$ Built-in text editor in Terminal Is there a built-in text editor in the terminal on Lion?  (Like, for example, nano on Linux.) How to add a Timeout to Console.ReadLine()? Prove that the natural map $\alpha : \text{Hom}(X,\text{Spec} A) \rightarrow \text{Hom}(A,\Gamma(X,\mathcal{O}_X))$ is an isomorphism This is question 2.4 in Hartshorne. Let $A$ be a ring and $(X,\mathcal{O}_X)$ a scheme. We have the associated map of sheaves $f^\#: \mathcal{O}_{\text{Spec } A} \rightarrow f_* \mathcal{O}_X$. Taking global sections  we obtain a homomorphism $A \rightarrow \Gamma(X,\mathcal{O}_X)$. Thus there is a natural map $\alpha : \text{Hom}(X,\text{Spec} A) \rightarrow \text{Hom}(A,\Gamma(X,\mathcal{O}_X))$. Show $\alpha$ is bijective.  I figure we need to start off with the fact that we can cover $X$ with affine open $U_i$, and that a homomorphism $A \rightarrow \Gamma(X,\mathcal{O}_X)$ induces a morphism of schemes from each $U_i$ to $\text{Spec} A$ and some how glue them together. But I have no idea how to show that the induced morphisms agree on intersections. How does this work? FFmpeg lossy compression problems How to typeset a primed vector I write a vector with an arrow above the letter using \vec{p} and sometimes the vector needs to be primed using ' (or ^{\prime}, equivalently).  However, neither  \vec{p}'     nor  \vec{p'}     looks satisfactory (first option: prime dash overlaps with vector arrow, second option: arrow looks misplaced).  How would one write a primed vector? Why do electrons move towards anode in the Photoelectric effect experiment? Consider a Photoelectric effect experiment apparatus as shown in Figure 11.1    The variation of photocurrent with the voltage applied across $\mathrm{A}$ and $\mathrm{C}$ is as shown in Figure 11.3.    My question –     Why is the current non-zero even if the voltage is zero?    Consider a situation in which the voltage across the plates is zero. Photons of sufficient frequency strike the outermost surface of the cathode, emitter plate $\mathrm{C}$. Electrons become free from the attraction of nucleus as their net energy becomes positive.  Why I think that there should be no net current?  As soon as photons strike the outermost surface of the cathode(assumed to be $1$ to $10$ atomic diameters thick), electrons become free. Now, as the electrons have kinetic energy, they'll come out of that outermost surface. Now, the electrons(with kinetic energy) should have an equal probability of going in every direction of the surface. So, around half of the electrons should go in the right direction towards the anode and strike it and half should go to the left, trying to penetrate the inner surfaces of the cathode plate.   Now, the electrons which will go to the right will face some field due to the space charge present in the tube but nevertheless, some electrons will definitely reach the anode plate. After they strike with the anode plate, they'll face some resistance but still, some electrons will reach the low resistance copper wires connected to the anode plate and milliammeter.  Now, consider the electrons which go towards left. They'll definitely face resistance from cathode plate but some will still reach the copper wire.  Now, as there is no potential difference applied between $\mathrm{A}$ and $\mathrm{C}$, there is no need of commutator and voltmeter. So, the ammeter is connected in series with the anode plates and cathode plates with copper wires.  Now, if there is steady state current in the circuit, electrons emitted to any one of the sides should dominate over the other. But the number of electrons emitted on both the sides is same, and all the electrons should face the same resistance as the current(electrons) should flow in a closed loop. But if the number and velocity of electrons coming from both the sides are same, there should be no net current. But there is. Why?  Edit –  From the answers of anna v and mmainville, it can be seen that the emission of electrons is dependent on the angle of emission of light.  But still, the question remains. If the electrons emitted towards anode have enough energy to complete the circuit and again reach the holes in cathode despite the resistance in between, the electrons emitted to the left should have enough energy to come to anode plate via copper wires and again get ejected through anode plate by the virtue of their kinetic energy and reach the cathode plate. In this case too, the current should be zero which is not true. Why? A local IP address (RFC 1918) on my network that is not … local I've noticed an IP address today on my network (192.168.15.1) which when visited actually goes to a page that has a router webconfig page (Not my router)  The puzzling thing is that i tried traceroute 192.168.15.1 which said that the distance between me and that host is 9 hops, whereas the distance between me and any other computer on my home network is 2 hops away (My computer --> Router --> Host)   Can someone please explain what's going on here ?  tracert output:  Tracing route to 192.168.15.1 over a maximum of 30 hops    1    &lt;1 ms    &lt;1 ms    &lt;1 ms  192.168.1.1   2    47 ms    51 ms    49 ms  [163.121.171.123]   3     *        *        *     Request timed out.   4   128 ms     *        *     163.121.211.134.dnet.net [163.121.211.134 ]   5     *      134 ms   126 ms  10.37.242.18   6    94 ms    90 ms    95 ms  10.35.2.157   7   178 ms   254 ms   206 ms  10.32.8.107   8    85 ms    57 ms   318 ms  10.35.0.186   9   215 ms   241 ms   230 ms  192.168.15.1  Trace complete. How to add a Login / Logout link in Main navigation menu? How can I add a Login link to the "Main navigation" menu?  When a user clicks on the link, it should take them to the /user/login page. Once they login, the link should change to say "Logout" instead of "Login".  Is there any way to add such a link? Why doesn't Ubuntu 16.04 in Windows have ubuntu.exe? What will happen with Blender 2.8 game engine? Blender has just purged the entire game engine from 2.8. There is no "Game Engine" mode anymore and there isn't even a logic editor. I tried installing a GitHub Blender release with the game engine mode working but it failed to install because all the game code was gutted. Does this mean that Blender 2.8 is going to lose the game engine, or is the game engine just being redone for better compatibility and whatnot? Implement numbering scheme like A,B,C... AA,AB,... AAA..., similar to converting a number to radix26 I want to implement numbering scheme like Microsoft Word uses for numbering.  first one gets = A,next is  B, next is C, .... then AA, then AB,....and so on.  as shown below   A        B  C  .   .  AA  AB  AC  .  .  AAA  AAB  ....   '=>' here means converted to.  some examples:       1 =&gt; A 26 =&gt; Z 27 =&gt; AA 52 =&gt; AZ 53 =&gt; BA   and heres the code for it:     var convertToNumberingScheme = function(n){         var x = n-1,             r26 = x.toString(26),             baseCharCode = "A".charCodeAt(0);          var arr = r26.split(''),             len = arr.length;          var newArr =arr.map(function(val,i){             val = parseInt(val,26);              if( (i === 0) &amp;&amp; ( len &gt; 1)){                 val = val-1;             }              return String.fromCharCode(baseCharCode + val);         });         return newArr.join('');     }   It seems to work fine, but any ideas if there are some potential bugs or ways to  optimize this. Why isn't Captain America worthy of Mjolnir? I haven't see the movie yet, but in the , we can see Cap trying to lift Mjolnir in the humorous scene where every Avenger gives it a try.  Mjolnir even moves just a tiny bit and at this very moment we see Thor's smile disappearing. This, of course, is intended as a humorous and lighthearted scene.  But it got me thinking: why isn't Captain America worthy of Mjolnir? He seems to be the most "perfect" human being anyone can be: brave, not selfish, legitimate leader, prompt to sacrifice (as seen in the first movie, when he jumps on a fake grenade).  I ask this question in light of this very specific scene, but I'm interested in the Marvel Universe (and Marvel Cinematic Universe) as a whole. Maybe Captain America did wield Mjolnir once? Maybe he has some individual flaw I am not aware of that makes him unworthy? Why is median age a better statistic than mean age? If you look at       Or this Wikipedia page      Clearly  seems to be the statistic of choice when it comes to ages.  I am not able to explain to myself why  would be a worse statistic. Why is it so?  Originally posted  because I did not know this site existed.