GATE CSE 2022 Q52: Properties of the adjacency matrix of a simple graph

Answer: A (official GATE 2022 final answer key)

Concept used: $(A^k)_{ij}$ counts walks of length exactly $k$; the sum of entries of $A$ is twice the number of edges.

First step: $(A^2)_{ii} = \sum_j a_{ij} a_{ji}$ counts neighbours of $i$; for the others try small counterexamples (a path on 3 vertices, a triangle plus an isolated vertex, two disjoint edges).