1. (i.) Justify the following statement: the total degree (sum of all degrees of all vertices)...

 

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1. (i.) Justify the following statement: the total degree (sum of all degrees of all vertices) of a finite graph is an isomorphism invariant. [10 marks] (ii.) Consider the multigraph below. (Note that this is not a weighted multigraph but has its edges labelled 1,2,..., 11.) a 1 b 2 3 4 6 0 7 8 s d 9 10 e 11 (a.) Decide whether or not this multigraph is Hamiltonian? Can you change this status by removing an edge? [5 marks] (b.) Decide whether or not this multigraph is Eulerian? Can you change this status by removing an edge? [5 marks] 1. (i.) Justify the following statement: the total degree (sum of all degrees of all vertices) of a finite graph is an isomorphism invariant. [10 marks] (ii.) Consider the multigraph below. (Note that this is not a weighted multigraph but has its edges labelled 1,2,..., 11.) a 1 b 2 3 4 6 0 7 8 s d 9 10 e 11 (a.) Decide whether or not this multigraph is Hamiltonian? Can you change this status by removing an edge? [5 marks] (b.) Decide whether or not this multigraph is Eulerian? Can you change this status by removing an edge? [5 marks]