Discrete Mathematics..

Trees 52912. (i) How many spanning trees does each of the following graphs have?(ii)What general observation can be made about the number ofspanning trees in a connected graph which has a bridge or a cutvertex? (‘Bridge’ is defined in exercise 10.11 above. A ‘cut vertex’in a connected graph is a vertex which, if removed from the graph,together with its incident edges, produces a disconnected graph.)13. In chemistry, graphs are used as symbolic representations of molecules.Each atom is represented as a vertex and each chemical bond isrepresented as an edge. For example, molecules of ethene (C 2 H 4 )andethanol (C 2 H 5 OH) can be represented by the following graphs.The alkane series of saturated hydrocarbons have chemical formulaC n H 2n+2 . Each carbon atom (C) always has valency (degree) 4 and eachhydrogen atom (H) always has valency (degree) 1.(i) Draw the graphs to represent methane (CH 4 ) and ethane (C 2 H 6 ).(ii) Prove that each alkane is represented by a tree. (Hint: usetheorem 10.6.)A structural isomer of a chemical compound is an isomorphism class ofits corresponding graph. This means that molecules of different structuralisomers of the same compound have non-isomorphic graphs.(iii) Butane (C 4 H 10 ) has two structural isomers; draw their graphs.(iv) Draw graphs of the different structural isomers of pentane (C 5 H 12 )and hexane (C 6 H 14 ).14. The alkenes and mono-cyclo-alkanes are unsaturated hydrocarbons withchemical formula C n H 2n , where n 2. (The graph of ethene, thesimplest alkene, is given above.)