Here are diagrams for four graphs, A, B, C and D.ghm.SnUCpPAkBCrWDC001 100 0 01 11(a) A graph E has the adjacenty matrix at right.100101This graph E is isomorphic to11 1010100A / B / C / D / none of these.H O110100101010 01 0(b) A graph F has the adjacenty matrix at right.100101This graph F is isomorphic to0 0101 1A / B / C / D / none of these1010 0101 10 0(c)
The graphs A and D are not isomorphic because:A has a vertex of degree 4, but D does not./ A has more vertices of degree 3 tahn does D./ A contains (as a subgraph) no circuit of length 6, but D does./ A contains (as a subgraph) a circuit of length 5, but D does not./ A contains (as a subgraph) a circuit of length 4, but D does not.(d) The graphs A and C are isomorphic.An isomorphism from A to C is given by a bijection9 : V(A) – V(C) for which o(a) = m and o(e) = n.It follows that o(d) = m / n / o / p / q / r(e) Another isomorphism from A to C is given by a bijectionp : V(A) – V(C) for which p(d) = q and p(e) = 0.It follows that p(a) = m / n / o / p / q / r(f) [Challenge] The number of isomorphisms from A to C is 3 / 4 / 6 / 9 / 12
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