GATE CS 2014 Set 1 — Question 61
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Engineering Mathematics → Graph Theory (Math) → Graph Terminology (Degree, Paths, Cycles)
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Question
Consider an undirected graph where self-loops are not allowed. The vertex set of is . There is an edge between and if and . The number of edges in this graph is __________.
Correct answer
506 to 506
Solution
The vertices are points on a grid. The condition and implies that two distinct vertices are connected if they are adjacent horizontally, vertically, or diagonally (King's moves in chess).Let . The total number of vertices is .
We calculate the sum of degrees of all vertices:
Sum = .
Each has 5 neighbors.
Sum = .
Each has 8 neighbors.
Sum = .Total sum of degrees = .By the Handshaking Lemma, the number of edges is half the sum of degrees:
.
We calculate the sum of degrees of all vertices:
1.Corner vertices: 4 corners.
Each has 3 neighbors.Sum = .
2.Edge vertices (excluding corners): on top/bottom + on left/right = .
For , vertices.Each has 5 neighbors.
Sum = .
3.Inner vertices: .
For , vertices.Each has 8 neighbors.
Sum = .Total sum of degrees = .By the Handshaking Lemma, the number of edges is half the sum of degrees:
.
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