Bounds for the Kirchhoff index via majorization techniques

Using a majorization technique that identifies the maximal and minimal vectors of a variety of subsets of ${mathbb{R}^{n}}$ , we find upper and lower bounds for the Kirchhoff index K(G) of an arbitrary simple connected graph G that improve those existing in the literature. Specifically we show that $$K(G) geq frac{n}{d_{1}} left[ frac{1}{1+frac{sigma}{sqrt{n-1}}} + frac{(n-2)^{2}}{n-1-frac{sigma}{sqrt{n-1}}}right] ,$$ where d ₁ is the largest degree among all vertices in G, $$sigma ^{2} = frac{2}{n} sum_{(i, j) in E} frac{1}{d_{i}d_{j}} = left( frac{2}{n}right) R_{-1}(G),$$ and R _?1(G) is the general Randi? index of G for ${alpha =-1}$ . Also we show that $$K(G) leq frac{n}{d_{n}}left( frac{n-k-2}{1-lambda _{2}}+frac{k}{2}+frac{1}{theta}right) ,$$ where d _n is the smallest degree, ${lambda _{2}}$ is the second eigenvalue of the transition probability of the random walk on G, $$k = left lfloor frac{lambda _{2} left( n-1right) +1}{lambda _{2}+1}rightrfloor {rm and}quadtheta = lambda _{2} left( n-k-2right) -k+2.$$