On \(K\)-eccentric and \(K\)-hyper eccentric indices of Benzenoid \(H_k\) system

Abstract

Let  \(\mathrm{G}\) be a connected graph with vertex set \(V(G)\) and edge set \(E(G)\). Bhanumathi and Easu Julia Rani introduced the first \(K\)-Eccentric index \(B_1 E(G)\) and the second \(K\) - Eccentric index \(B_2 E(G)\) of a graph \(G\) as \(B_1 E(G)=\Sigma_{u e}\left[e_G(u)+e_{L(G)}(e)\right], B_2 E(G)=\sum_{u e}\left[e_G(u) e_{L(G)}(e)\right]\). They also defined the first \(K\)-Hyper eccentric index \(H B_1 E(G)\) and the second \(K\)-Hyper eccentric index \(H B_2 E(G)\) of a graph \(G\) as \(H B_1 E(G)=\sum_{u e}\left[e_G(u)+e_{L(G)}(e)\right]^2, H B_2 E(G)=\sum_{u e}\left[e_G(u) e_{L(G)}(e)\right]^2\) where in all the cases \(u e\) means that the vertex \(u\) and edge \(e\) are incident in \(G\) and \(e_{L(G)}(e)\) is the eccentricity of \(\mathrm{e}\) in the line graph \(L(G)\) of \(G\). They have defined the multiplicative version of these indices also. In this paper, we calculate the first and second \(K\) eccentric, the first and second K-hyper eccentric indices and their multiplicative versions of benzenoid \(H_k\) system.