Question

eight students form a math homework group. the students in the group are joe, earl, zippy, annette, stinky, muriel, chet, and oswaldo. prior to forming the group, stinky was friends with everyone but muriel. muriel was friends with zippy, annette, chet, and earl. joe was friends with stinky, earl, oswaldo, and zippy. letting an edge between two students represent friendship, which of the following graphs models the friendships among these students? which of the graphs models the the friendships? \bigcirc graph i \bigcirc graph ii

Explanation:

Step1: Analyze Stinky's friendships

Stinky was friends with everyone but Muriel. So there should be edges from Stinky (\(S\)) to Joe (\(J\)), Earl (\(E\)), Zippy (\(Z\)), Annette (\(A\)), Chet (\(C\)), and Oswaldo (\(O\)) but no edge between \(S\) and \(M\) (Muriel).

Step2: Analyze Muriel's friendships

Muriel was friends with Zippy (\(Z\)), Annette (\(A\)), Chet (\(C\)), and Earl (\(E\)). So there are edges \(M - Z\), \(M - A\), \(M - C\), \(M - E\).

Step3: Analyze Joe's friendships

Joe was friends with Stinky (\(S\)), Earl (\(E\)), Oswaldo (\(O\)), and Zippy (\(Z\)). So there are edges \(J - S\), \(J - E\), \(J - O\), \(J - Z\).

In Graph I:

• Check Stinky's connections: \(S\) is connected to \(J\), \(Z\), \(A\), \(E\), \(O\), \(C\) (correct as \(S\) is not connected to \(M\)).
• Check Muriel's connections: \(M\) is connected to \(Z\), \(A\), \(C\), \(E\) (correct).
• Check Joe's connections: \(J\) is connected to \(S\), \(Z\), \(E\), \(O\) (correct).

In Graph II:

• The position - labeling of \(J\) (Joe) and \(Z\) (Zippy) is swapped. Since in the problem description, we have a set of relationships based on the original labels (e.g., Joe's friends are based on his label \(J\) in the context of the problem's naming of students), the incorrect labeling of vertices (swapping \(J\) and \(Z\)) makes Graph II wrong.

Answer:

Graph I