1. **Problem Statement:** Find the number of vertices, number of edges, degree of each vertex, and identify isolated and pendant vertices for the graphs G_1, G_2, and G_3. 2. **Definitions and Rules:** - Number of vertices: count of distinct vertices. - Number of edges: count of distinct edges; loops count as one edge. - Degree of a vertex: number of edges incident to it; loops count twice. - Isolated vertex: vertex with degree 0. - Pendant vertex: vertex with degree 1. --- ### Graph G_1 Vertices: a, b, c, d, e, f (6 vertices) Edges: (a-f), (a-b), (b-c), (b-e), (e-f), (b-f) (6 edges) Degrees: - $\deg(a) = 2$ (edges a-f, a-b) - $\deg(b) = 4$ (edges a-b, b-c, b-e, b-f) - $\deg(c) = 1$ (edge b-c) - $\deg(d) = 0$ (isolated) - $\deg(e) = 2$ (edges b-e, e-f) - $\deg(f) = 2$ (edges a-f, e-f, plus b-f actually makes 3 edges incident to f, so correct degree is 3) Correction for $\deg(f)$: Edges incident to f: a-f, e-f, b-f So $\deg(f) = 3$ Isolated vertices: d Pendant vertices (degree 1): c --- ### Graph G_2 Vertices: a, b, c, d, e, f (6 vertices) Edges: (a-b), loop on a, (b-d), (d-c), loop on c, (e-a), (e-f), (b-e), (a-e) Count edges: - a-b - loop a - b-d - d-c - loop c - e-a - e-f - b-e - a-e (note a-e appears twice, count once) Edges count: 8 distinct edges (loops count as edges) Degrees (loops count twice): - $\deg(a) = $ edges a-b, a-e (twice?), loop on a counts 2 Edges incident to a: a-b, a-e (twice?), loop a Note: a-e appears twice, so two edges between a and e So edges incident to a: a-b, a-e (twice), loop a Degree $= 1 (a-b) + 2 (a-e twice) + 2 (loop) = 5$ - $\deg(b) = $ edges a-b, b-d, b-e Edges incident to b: a-b, b-d, b-e Degree $= 3$ - $\deg(c) = $ edges d-c, loop c Loop counts twice Degree $= 1 (d-c) + 2 (loop) = 3$ - $\deg(d) = $ edges b-d, d-c Degree $= 2$ - $\deg(e) = $ edges e-a (twice), e-f, b-e Edges incident to e: a-e (twice), e-f, b-e Degree $= 4$ - $\deg(f) = $ edge e-f Degree $= 1$ Isolated vertices: none Pendant vertices (degree 1): f --- ### Graph G_3 Vertices: a, b, c, d, e, f, g, h, i (9 vertices) Edges: (a-i), (a-f), (a-b), (b-h), (c-g), (b-g), (c-e), (g-e), (c-b), (a-c) Count edges: 10 Degrees: - $\deg(a) = $ edges a-i, a-f, a-b, a-c Degree $= 4$ - $\deg(b) = $ edges a-b, b-h, b-g, c-b Degree $= 4$ - $\deg(c) = $ edges c-g, c-e, c-b, a-c Degree $= 4$ - $\deg(d) = 0$ (isolated) - $\deg(e) = $ edges c-e, g-e Degree $= 2$ - $\deg(f) = 0$ (isolated) - $\deg(g) = $ edges c-g, b-g, g-e Degree $= 3$ - $\deg(h) = $ edge b-h Degree $= 1$ - $\deg(i) = $ edge a-i Degree $= 1$ Isolated vertices: d, f Pendant vertices: h, i --- **Final answers:** - G_1: vertices=6, edges=6, degrees: a=2, b=4, c=1, d=0, e=2, f=3; isolated: d; pendant: c - G_2: vertices=6, edges=8, degrees: a=5, b=3, c=3, d=2, e=4, f=1; isolated: none; pendant: f - G_3: vertices=9, edges=10, degrees: a=4, b=4, c=4, d=0, e=2, f=0, g=3, h=1, i=1; isolated: d, f; pendant: h, i