Network in Graph Theory · Form 4
Vertices, edges and degree, explained
A network is just dots joined by lines: the dots are vertices, the lines are edges, and a vertex's degree counts how many edge-ends meet there.
The three words, side by side
A vertex is a single point in the network, it stands for a thing, like a town, a person or a computer. An edge is a line joining two vertices, showing they are connected in some way.
The degree of a vertex is simply how many edge-ends meet at it, so a point with three lines coming out of it has degree 3.
Why degree is worth counting
Degree measures how connected a single point is, and adding up every degree in a network always gives exactly twice the number of edges, because each edge has two ends. That fact makes a quick check: if your degrees add to an odd number, you have miscounted somewhere.
A vertex with degree 0 is isolated, joined to nothing.
Where students slip
The commonest error is treating a crossing of two lines as a vertex, but only a marked dot counts; lines may simply pass over each other. A loop, an edge that starts and ends at the same vertex, adds 2 to that vertex's degree, not 1, since both its ends are attached there.
Draw the network with space so edges drawn close together are not miscounted.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I count the degree of a vertex when a loop is involved?
Count every edge-end touching the vertex. A normal edge connecting two different vertices adds 1 to each vertex's degree, but a loop, an edge starting and ending at the same vertex, adds 2 to that vertex's degree, since both ends meet there.
Is there a quick way to check my degree count is right?
The sum of all vertices' degrees always equals twice the number of edges, since every edge contributes exactly two edge-ends. Use this to verify a network diagram you've drawn, or to find the number of edges when only the degrees are given.
What mistake do students commonly make when counting degree?
Counting a loop as only 1 towards degree, or forgetting to count it at all. Remember a loop contributes 2 to its vertex's degree, and check whether the question's network allows loops or multiple edges before you start counting.