Prove that in any graph with more than one vertex, there must exist two vertices of the same degree. [Hint:Pigeon-Hole Principle.]...

Let $G$ be a simple graph. Show that the relation $R$ on the set of vertices of $G$ such that $u R v$ if and only if there is an edge associated to $\{u, v\}$ is a symmetric, irreflexive relation on $G .$

In this question. We are presented with a craft D that has a loop on Everybody is's and we defy a relation r b such that on the scent off. What pisses off G, what is you? And we will be related if and believe there is an age between them and we asked to show that our is reflexive and symmetry relation. And it's very simple, so reflexive any vortices We're related to itself because off the loop, since we defy d as having a loop on every everyone, this is next Symmetry probably is also simple. If we have you relate to be in there, isn't it joining you and be right. And that would also mean that isn't joining. We end you and so we have you know we relate to you. So it is symmetry that is it for this video. Thank you

High in this question. We are asked to show that a symbol crap with at least two witnesses must contain, uh to what? This is a pal them that help the same degree. So this may be the same as these two or not, and doesn't. It doesn't force to be the same. It's just that if the graft is at least to what this is all more, we can fire a pair with same degree and to prove this, we will use proved by contradiction. So we supposed they're really exists. A symbol grab with no Powell witnesses off the same degree. Okay, so, everyone, this is how different degrees. Now let's look at this crap. It is finite. We all there would finally grow up. So it has n witnesses, right? When a craft have been witnesses ish one of them who have possible degree ranging from zero to in minus one zero. When justice is isolate that and in minus one, when it has ages to every other witnesses, right, it can. They were have and degrees in a simple graph. Okay, now, when we assume that all what this is must have different degrees, it means that So everyone, is it blue shoes? One off these in choice and the ones one was chosen. The other. What is it? Can choose it again. Right. So the degree sequins off this graph would be just everything here. So from in mice one ranging to zero, each number got chosen ones. And this leads to a contradiction because what this is with in minus one degree and Syria, the key can both exits in a grab with n witnesses. Why? Well, we have zero degree, which is isolated, right? That's called you and we have in minus one degree. This could be which I just say that it must have. It means that it has edges to every other. What? This is in the graph Because we only have in what this is. And and so one off those edges must connect to you. But then it will make you on degree one, not zero. So it's it can happen. Born off these what does this can co exist? Can board exist. So this is a contradiction that mean the assumption that we start with the they're really exist. Sachin, a steamboat grab is wrong. So there's no such thing. Bagram And so we have prevent a statement. Thank you

Indiscretion. We are asked to show that for any positive intelligence in and we have a complete graph kin, Any sub graph induced by in on If he's upset off what this is off this can will be a complete grab as well. So we have, for example, Careful. We shoes some witnesses off it. And in news, a sub crab off those. What is you're gonna get something like this and we have to show that you always end up being a complete graph as well. And let's go. So I suppose not Suppose we have the complete garb, kid. So we have in witnesses, right? And it is complete. So every possible edges Are you kidding? So any pair off the I V day, we have a choice between them and we suppose we have a substance. A we rename element India as you want to you em So these gonna be those we i with this is we don't know which in the scent, eh? But it doesn't matter my by taking this from a complete grab. It means that in and induce up crab that we are creating with a any pair over disease. You are you j who have an age between them as well, right? Because they they have this age in the already know graph. And so a grab created by a ass politics. And I guess this is set B be asked edges will be will be complete because ah, specifically, it would be I some coffee too, kay em because it has em were ticks says, um any to what? This is house aged between them. Right? That's the definition. Off came. And so this in news sub craft will be a complete graph. Always. And that's how you show it, huh? Thank you.

So for this direction, if the graph has last circuit, then it is obviously it is with connected weekly. And that and if the circuit is a, you know, a one adult in a zero we can find a zero with ample You I So look, I hear Yeah, if we have this in degree is one and we have our degree Also, one said in degree is ours equal to Diggory. So for this direction, if the graph is waking, inactive and in degree is our degree then are you things at rhythm one? In the test book, we can construct well circuit just to replace it is in the factory them bye directed itis and we can't train and circuit just use algorithm My third of rhythm and replace that is by directed I just