Question 26

Let us examine each statement individually:

1. (P) The line graph of a cycle is a cycle:
   A cycle $C_n$ on $n$ vertices has $n$ edges. Each edge shares an endpoint with exactly two other edges. Therefore, every vertex in $L(C_n)$ has degree 2, and the connectivity of $C_n$ implies $L(C_n)$ is connected. A connected 2-regular graph is a cycle $C_n$. Thus, (P) is TRUE.

2. (Q) The line graph of a clique is a clique:
   Consider the complete graph $K_4$, which is a clique on 4 vertices with 6 edges. In $K_4$, there exist pairs of disjoint edges (for example, $(1,2)$ and $(3,4)$) that share no common vertex. In $L(K_4)$, these two vertices will not be connected by an edge. Therefore, $L(K_4)$ is not a clique (it is the octahedral graph). Thus, (Q) is FALSE.

3. (R) The line graph of a planar graph is planar:
   Consider the star graph $K_{1,5}$, which has a central vertex connected to 5 leaves. $K_{1,5}$ is a tree, hence planar. In its line graph $L(K_{1,5})$, every pair of edges is incident at the central vertex, so all 5 edges are mutually adjacent. Thus, $L(K_{1,5}) = K_5$, which is non-planar by Kuratowski's theorem. Thus, (R) is FALSE.

4. (S) The line graph of a tree is a tree:
   Consider the star graph $K_{1,3}$, which is a tree. Its line graph $L(K_{1,3})$ is the complete graph $K_3$ (a 3-cycle), which contains a cycle and is therefore not a tree. Thus, (S) is FALSE.

Since only statement (P) is true, option (A) is correct.