What are Adjacent Vertices Graph Theory

hey everyone welcome to wrath of math I'm your host Sean Ian's in today's video we'll be answering the question what are adjacent vertices so let's get right into it I can't swipe this blasted page okay here we go so let's say we've got two vertices one and two cliche names but they're effective right now one and two are not adjacent but why is that well you are here to find the answer they're not adjacent because there is no edge joining them but let's say we draw an edge drawing this edge in blue this edge joins one and two they are now adjacent vertices because there is an edge joining them so let's look at the non adjacent and the adjacent edges side by side just to make sure it's absolutely clear so up top we've got two adjacent vertices one and two again they're joined by this blue edge so they are adjacent and then at the bottom the vertices 3 & 4 are non adjacent because there is no edge joining them and then let's take a look at some adjacent and non-adjacent vertices in a slightly more interesting graph here of course we've got a graph G and we've got five vertices and a few edges so can you tell which vertices are and are not adjacent well the adjacent ones of course are the ones with edges joining them and I've taken the liberty of labeling the edges so we can see that one in two are adjacent vertices in this graph because they're joined by the edge a one in four are jacent vertices because they're joined by the edge B and two and three are adjacent vertices because they are nicely joined by that edge C now some of the non adjacent vertices of course we can see I'll Circle them with let's say this green color one and three are non adjacent we see there's no edge between them same with two and four they are not adjacent because there's no edge joining them either also five over here is not adjacent to any vertex thus we can say that five is an isolated vertex and again that's because it is not adjacent to any other vertices that means it's an isolated vertex now let's move on to the formal definition just take a quick look at this before we go so the definition let G be a graph and a and B are vertices in G then a and B are adjacent in G if and only if the set containing a and B is an element of the edge set of the graph G this just says that there is an edge in G joining a and B and I should also mention that the edge set of G that is a set that consists of two element subsets of the vertex set like that but edges are often written just like this instead simply because it's easier and it conveys the same information that it's an edge joining a and B and so that's the definition of adjacent vertices two vertices are adjacent in a graph if there is an edge in that graph joining those vertices and that's really all there is to it so I hope you enjoyed the video I hope it helped you understand what adjacent vertices are let me know in the comments if you have any questions need anything clarified or have any other video requests thank you very much for watching I'll see you next time and be sure to subscribe to the swankiest math videos on the internet [Music] you

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