Let G be a connected graph of order n and size n. Prove that G contains a single cycle.
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- Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and edge e of G, v and e lie on a common cycle of G.Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only if u and v are in the same connected component of G.
- Suppose a graph G has the following Harmiltonian Cycle: BAGDFHEJKXB a. How many vertices does G have? b. What is the length of the cycle? C Find two distinct paths between D'and X (distinct means they use different edges) Type your answer in the space provided. For the toolbar, press ALT+F10 (PC) or ALT +FN+F10 (Mac).Let G be a C7 graph (a circuit graph with 7 vertices). Prove that Gc has a Euler cycle.Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph obtained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) 4e)}.