(a) 7 edges in a connected graph with 7 even vertices and no odd vertices.
(b) 16 vertices in a connected graph with 6 even and 2 odd vertices.
(c) Minimum 18 edges in a connected graph with 10 even and 8 odd vertices.
Let's analyze each question:
(a) For a connected graph with no odd vertices and 7 even vertices, how many edges does the graph have?
The answer is given by Euler's formula, which states that for a connected graph with
vertices,
edges, and
faces (including the exterior), the formula is
In this case, there are no odd vertices, so all vertices have even degrees. Let
be the degree of each even vertex. The sum of the degrees is
(since each edge contributes 2 to the sum of degrees). Since there are no odd vertices, the sum of degrees must be even. Therefore, \(e\) (the total number of edges) must be 7. So, the answer to (a) is 7.
(b) In a connected graph with 6 even vertices and 2 odd vertices, what is the total number of vertices?
Each edge contributes 2 to the sum of degrees. The sum of degrees for the even vertices is
, and for the odd vertices, it's
The total sum of degrees is
. Since each vertex contributes 1 to the sum of degrees, there are \(16\) vertices in total. So, the answer to (b) is 16.
(c) In a connected graph with 10 even vertices and 8 odd vertices, what is the minimum number of edges the graph can have?
Similar to (a), we use Euler's formula. The sum of the degrees for even vertices is
, and for odd vertices, it's
. The total sum of degrees is
The number of edges
is half of the sum of degrees, so
Therefore, the answer to (c) is 18.
The probable question maybe:
(a) For a connected graph with no odd vertices and 7 even vertices, how many edges does the graph have?
(b) In a connected graph with 6 even vertices and 2 odd vertices, what is the total number of vertices?
(c) In a connected graph with 10 even vertices and 8 odd vertices, what is the minimum number of edges the graph can have?