Graph Theory

Step 1: is Eulerian if all vertices have even degree. In , degree of each vertex = .}
Step 2: is even when is odd. So is Eulerian for odd . Option (A) is correct.}

Step 1: Let = number of degree 1 vertices, = number of degree 3 vertices. Then total vertices .}
Step 2: Sum of degrees = . Also . Solve: . So . For tree, need integer solution. gives . So pendant vertices = .}

Step 1: Wheel has a central vertex connected to all vertices of a cycle .}
Step 2: For , the cycle is odd when is odd, i.e., even. But the central vertex creates odd cycles. So is not bipartite for any .}

Step 1: Understanding the Concept:
A graph is connected if there is a path between any two vertices.
Step 2: Detailed Explanation:
The graph shown has all vertices connected to each other through some path. There is no isolated vertex or separate component.
Step 3: Final Answer:
Connected.

Step 1: Understanding the Concept:
Vertex connectivity is the minimum number of vertices whose removal disconnects the graph.
Step 2: Detailed Explanation:
A tree has at least one leaf (vertex of degree 1). Removing a leaf does not disconnect the tree. However, removing a cut vertex (articulation point) disconnects it. The vertex connectivity of any tree is 1.
Step 3: Final Answer:
One.

Concept:

Step 1: Possible values.


Step 2: Check condition.
Graph with more than one cycle degree Valid case: