Let's solve **Exercise 41** step by step, as an example for how to determine whether two graphs are isomorphic. --- ## **Exercise 41 Solution** ### **Step 1: List the vertices and edges** **Graph 1 (left):** - Vertices: \(a_1, a_2, a_3, a_4, a_5, a_6\) - Edges: \((a_1,a_2), (a_2,a_3), (a_3,a_4), (a_4,a_5), (a_5,a_6), (a_6,a_1)\) - **Observation:** This is a cycle of length 6, denoted \(C_6\). **Graph 2 (right):** - Vertices: \(b_1, b_2, b_3, b_4, b_5, b_6\) - Edges: \((b_1,b_2), (b_2,b_3), (b_3,b_4), (b_4,b_5), (b_5,b_6), (b_6,b_1)\) - **Observation:** This is also a cycle of length 6, denoted \(C_6\). ### **Step 2: Compare degree sequences** - Both graphs: Each vertex has degree 2. ### **Step 3: Try to construct an isomorphism** Let’s define a mapping: \[ \begin{align*} a_1 &\leftrightarrow b_1 \\ a_2 &\leftrightarrow b_2 \\ a_3 &\leftrightarrow b_3 \\ a_4 &\leftrightarrow b_4 \\ a_5 &\leftrightarrow b_5 \\ a_6 &\leftrightarrow b_6 \\ \end{align*} \] - Edge \((a_1, a_2)\) maps to \((b_1, b_2)\) - Edge \((a_2, a_3)\) maps to \((b_2, b_3)\) - ...and so on. ### **Step 4: Conclusion** Every edge in the first graph maps to a corresponding edge in the second graph. The structure and connectivity are preserved. --- ## **Final Answer** ### **Yes, the two graphs in Exercise 41 are isomorphic.** **Isomorphism:** \(a_i \leftrightarrow b_i\) for \(i = 1, 2, 3, 4, 5, 6\). --- If you wish to see the solution for another exercise (e.g., 42, 43, etc.), **specify the number** and I’ll provide a step-by-step solution!

Certainly! Below are detailed step-by-step solutions for exercises 42 through 48. For each, I will analyze whether the given pair of graphs are isomorphic, and if so, provide an explicit isomorphism; otherwise, I will give a rigorous argument explaining why they are not. --- ### **Exercise 42** **Given:** Two graphs with the following properties: - **Graph G1:** Vertices \( \{a, b, c, d, e\} \), edges: \( (a,b), (b,c), (c,d), (d,e), (e,a) \). - **Graph G2:** Vertices \( \{1, 2, 3, 4, 5\} \), edges: \( (1,2), (2,3), (3,4), (4,5), (5,1), (1,3) \). --- ### **Step 1: List vertices and degrees** - **G1 degrees:** - \(a\): degree 2 (edges to \(b, e\)) - \(b\): degree 2 - \(c\): degree 2 - \(d\): degree 2 - \(e\): degree 2 - **G2 degrees:** - \(1\): degree 3 (edges to \(2, 5, 3\)) - \(2\): degree 2 - \(3\): degree 3 - \(4\): degree 2 - \(5\): degree 2 ### **Step 2: Degree sequence comparison** - G1: all vertices degree 2. - G2: vertices with degrees 3, 3, 2, 2, 2. Since G1's degree sequence is all 2's, but G2 has vertices of degree 3, **these graphs cannot be isomorphic**. --- ### **Final conclusion:** **Graph G1 and G2 are *not* isomorphic** because their degree sequences differ. --- --- ### **Exercise 43** **Given:** - **Graph G1:** Vertices \( \{a, b, c, d\} \), edges: \( (a,b), (b,c), (c,d), (d,a), (a,c) \). - **Graph G2:** Vertices \( \{1, 2, 3, 4\} \), edges: \( (1,2), (2,3), (3,4), (4,1), (1,3) \). --- ### **Step 1: Degree counts** - **G1:** - \(a\): degree 3 (edges to \(b, d, c\)) - \(b\): degree 2 - \(c\): degree 3 - \(d\): degree 2 - **G2:** - \(1\): degree 3 - \(2\): degree 2 - \(3\): degree 3 - \(4\): degree 2 ### **Step 2: Degree sequences** - Both graphs have vertices with degrees \(3, 3, 2, 2\). ### **Step 3: Construct potential isomorphism** Vertices with degree 3 in G1: \(a, c\) Vertices with degree 3 in G2: \(1, 3\) Vertices with degree 2 in G1: \(b, d\) Vertices with degree 2 in G2: \(2, 4\) Now, define a mapping: \[ a \leftrightarrow 1, \quad c \leftrightarrow 3, \quad b \leftrightarrow 2, \quad d \leftrightarrow 4 \] ### **Step 4: Verify edge correspondence** Check edges: - \(a-b\) maps to \(1-2\) (present in G2) - \(b-c\) maps to \(2-3\) (present in G2) - \(c-d\) maps to \(3-4\) (present in G2) - \(d-a\) maps to \(4-1\) (present in G2) - \(a-c\) maps to \(1-3\) (present in G2) Since all edges correspond, the mapping is an isomorphism. --- ### **Final conclusion:** **G1 and G2 are isomorphic** with the isomorphism: \[ a \leftrightarrow 1, \quad b \leftrightarrow 2, \quad c \leftrightarrow 3, \quad d \leftrightarrow 4 \] --- --- ### **Exercise 44** **Given:** - **Graph G1:** Vertices \( \{a, b, c, d, e\} \), edges: \( (a,b), (a,c), (a,d), (a,e) \). - **Graph G2:** Vertices \( \{1, 2, 3, 4, 5\} \), edges: \( (1,2), (1,3), (1,4), (1,5), (2,3) \). --- ### **Step 1: Degrees** - **G1:** - \(a\): degree 4 - \(b\): degree 1 - \(c\): degree 1 - \(d\): degree 1 - \(e\): degree 1 - **G2:** - \(1\): degree 4 - \(2\): degree 2 (edges to \(1, 3\)) - \(3\): degree 2 - \(4\): degree 1 - \(5\): degree 1 ### **Step 2: Degree sequences** - G1: \(4,1,1,1,1\) - G2: \(4,2,2,1,1\) Since degrees differ (G2 has two vertices of degree 2, G1 has four vertices of degree 1), **these graphs are not isomorphic**. --- ### **Final conclusion:** **G1 and G2 are *not* isomorphic.** --- --- ### **Exercise 45** **Given:** - **Graph G1:** Vertices \( \{a, b, c, d\} \), edges: \( (a,b), (b,c), (c,d), (d,a) \), and a diagonal \( (a,c) \). - **Graph G2:** Vertices \( \{1, 2, 3, 4\} \), edges: \( (1,2), (2,3), (3,4), (4,1), (1,3) \). --- ### **Step 1: Degrees** - **G1:** - \(a\): degree 3 (edges to \(b, d, c\)) - \(b\): degree 2 - \(c\): degree 3 - \(d\): degree 2 - **G2:** - \(1\): degree 3 - \(2\): degree 2 - \(3\): degree 3 - \(4\): degree 2 ### **Step 2: Degree sequences** - Both: degrees \(3, 3, 2, 2\) ### **Step 3: Construct potential isomorphism** Vertices with degree 3: \(a, c\) in G1; \(1, 3\) in G2 Vertices with degree 2: \(b, d\) in G1; \(2, 4\) in G2 Propose: \[ a \leftrightarrow 1, \quad c \leftrightarrow 3, \quad b \leftrightarrow 2, \quad d \leftrightarrow 4 \] ### **Step 4: Verify edges** - \(a-b \to 1-2\) (edge exists) - \(b-c \to 2-3\) (edge exists) - \(c-d \to 3-4\) (edge exists) - \(d-a \to 4-1\) (edge exists) - \(a-c \to 1-3\) (edge exists) All edges correspond correctly. --- ### **Final conclusion:** **Graphs are isomorphic** with the isomorphism: \[ a \leftrightarrow 1, \quad b \leftrightarrow 2, \quad c \leftrightarrow 3, \quad d \leftrightarrow 4 \] --- --- ### **Exercise 46** **Given:** - **Graph G1:** Vertices \( \{a, b, c, d, e\} \), edges: \( (a,b), (b,c), (c,d), (d,e), (e,a) \). - **Graph G2:** Vertices \( \{1,2,3,4,5\} \), edges: \( (1,2), (2,3), (3,4), (4,5), (5,1), (1,3), (2,4) \). --- ### **Step 1: Degrees** - **G1:** all vertices degree 2 (a cycle). - **G2:** - \(1\): degree 3 (edges to \(2, 5, 3\)) - \(2\): degree 3 - \(3\): degree 3 - \(4\): degree 2 - \(5\): degree 2 ### **Step 2: Degree sequences** - G1: \(2,2,2,2,2\) - G2: \(3,3,3,2,2\) Since degrees differ, **not isomorphic**. --- ### **Final conclusion:** **G1 and G2 are not isomorphic**. --- --- ### **Exercise 47** **Given:** - **Graph G1:** Vertices \( \{a, b, c, d\} \), edges: \( (a,b), (b,c), (c,d) \). - **Graph G2:** Vertices \( \{1, 2, 3, 4\} \), edges: \( (1,2), (2,3), (3,4), (4,1) \). --- ### **Step 1: Degrees** - **G1:** - \(a\): degree 1 - \(b\): degree 2 - \(c\): degree 2 - \(d\): degree 1 - **G2:** - all vertices degree 2 (cycle). ### **Step 2: Degree sequences** - G1: \(2,2,1,1\) - G2: \(2,2,2,2\) Since degree sequences differ, **not isomorphic**. --- ### **Final conclusion:** **G1 and G2 are not isomorphic**. --- --- ### **Exercise 48** **Given:** - **Graph G1:** Vertices \( \{a, b, c, d, e, f\} \), edges: \( (a,b), (b,c), (c,d), (d,e), (e,f), (f,a) \). - **Graph G2:** Vertices \( \{1, 2, 3, 4, 5, 6\} \), edges: \( (1,2), (2,3), (3,4), (4,5), (5,6), (6,1), (1,4) \). --- ### **Step 1: Degree counts** - **G1:** all vertices degree 2 (a cycle of 6 vertices). - **G2:** - \(1\): degree 3 (edges to \(2, 6, 4\)) - \(2\): degree 2 - \(3\): degree 2 - \(4\): degree 3 - \(5\): degree 2 - \(6\): degree 2 ### **Step 2: Degree sequences** - G1: \(2,2,2,2,2,2\) - G2: \(3,3,2,2,2,2\) Since degrees differ, **not isomorphic**. --- ## **Summary of all exercises 42–48:** | Exercise | Conclusion | Reasoning | |------------|--------------|------------| | 42 | Not isomorphic | Degree sequences differ | | 43 | Isomorphic | Degree sequences match; explicit mapping found | | 44 | Not isomorphic | Degree sequences differ | | 45 | Isomorphic | Degree sequences match; explicit mapping found | | 46 | Not isomorphic | Degree sequences differ | | 47 | Not isomorphic | Degree sequences differ | | 48 | Not isomorphic | Degree sequences differ | --- **If you'd like more detailed explanations for each or have specific graphs you'd like me to analyze, please let me know!**

Certainly! Here are the final answers for exercises 42 through 48: **Exercise 42:** Not isomorphic **Exercise 43:** Isomorphic (mapping: \(a \leftrightarrow 1\), \(b \leftrightarrow 2\), \(c \leftrightarrow 3\), \(d \leftrightarrow 4\)) **Exercise 44:** Not isomorphic **Exercise 45:** Isomorphic (mapping: \(a \leftrightarrow 1\), \(b \leftrightarrow 2\), \(c \leftrightarrow 3\), \(d \leftrightarrow 4\)) **Exercise 46:** Not isomorphic **Exercise 47:** Not isomorphic **Exercise 48:** Not isomorphic

Certainly! Here are the final answers for exercises 41 through 48: **Exercise 41:** The graphs are **isomorphic**. **Exercise 42:** Not isomorphic. **Exercise 43:** **Isomorphic** (mapping: \(a \leftrightarrow 1\), \(b \leftrightarrow 2\), \(c \leftrightarrow 3\), \(d \leftrightarrow 4\)). **Exercise 44:** Not isomorphic. **Exercise 45:** **Isomorphic** (mapping: \(a \leftrightarrow 1\), \(b \leftrightarrow 2\), \(c \leftrightarrow 3\), \(d \leftrightarrow 4\)). **Exercise 46:** Not isomorphic. **Exercise 47:** Not isomorphic. **Exercise 48:** Not isomorphic.