Let G be a group order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?

Let G = (V, E) be an undirected unweighted connected graph. The diameter of G is defined as:

diam(G) = $$\displaystyle\max_{u, x\in V}$$ {the length of shortest path between u and v}

Let M be the adjacency matrix of G.

Define graph G₂ on the same set of vertices with adjacency matrix N, where

$$N_{ij} =\left\{ {\begin{array}{*{20}{c}} {1 \ \ \text{if} \ \ {M_{ij}} > 0 \ \ \text{or} \ \ P_{ij} > 0, \ \text{where} \ \ P = {M^2}}\\ {0, \ \ \ \ \ \text{otherwise}} \end{array}} \right.$$

Which one of the following statements is true?

Graph G is obtained by adding vertex s to K_3,4 and making s adjacent to every vertex of K_3,4. The minimum number of colours required to edge-colour G is _____.

Suppose L = { p, q, r, s, t } is a lattice represented by the following Hasse diagram: For any $$x, y ∈ L$$, not necessarily distinct, $$x ∨ y$$ and x ∧ y are join and meet of x, y, respectively. Let $$L^3 = \left\{\left(x, y, z\right): x, y, z ∈ L\right\}$$ be the set of all ordered triplets of the elements of L. Let p_r be the probability that an element $$\left(x, y,z\right) ∈ L^3$$ chosen equiprobably satisfies $$x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)$$. Then