Table 2 Strength-weighted values of \(G^*\).

From: Distance based topological characterization, graph energy prediction, and NMR patterns of benzene ring embedded in P-type surface in 2D network

Quotient graph

Vertex weight : \(w_v\)

Vertex strength : \(v_s\)

\(G/Z_{1i}\)

\(i \in \mathbb {N}_{m-1}\)

\(x_1 = 12i^2 - 2i\)

\(x_2 = |V(G)| - x_1 - 4i\)

\(y_1 = 16i^2 - 6i\)

\(y_2 = |E(G)| - y_1 - 8i\)

\(G/Z_{2i}\)

\(i \in \mathbb {N}_{n-m+1}\)

\(x_3 = 24mi +12m^2 -26m\)

\(x_4 = |V(G)| - x_3 - 4m\)

\(y_3 = 32mi + 16m^2 - 38m - 2i+2\)

\(y_4 = |E(G)| - y_3 - 8m\)

\(G/O_{1i}\)

\(i \in \mathbb {N}_{m}\)

\(x_5 = 12i^2 - 12i +3\)

\(x_6 = |V(G)| - x_5\)

\(y_5 = 16i^2 - 20i + 6\)

\(y_6 = |E(G)| - y_5 - (4i-2)\)

\(G/O_{2i}\)

\(i \in \mathbb {N}_{n-m}\)

\(x_7 = 24mi +12m^2 -12m\)

\(x_8 = |V(G)| - x_7\)

\(y_7 = 32mi + 16m^2 - 20m - 2i +1\)

\(y_8 = |E(G)| - y_7 - 4m\)

\(G/V_i\)

\(i \in \mathbb {N}_{2n-1}\)

\(x_9 = 12mi\)

\(x_{10} = |V(G)| - x_9\)

\(y_9 = 16mi - 2m - i\)

\(y_{10} = |E(G)| - y_9 - 2m\)

\(G/H_i\)

\(i \in \mathbb {N}_{2m-1}\)

\(x_{11} = 12ni\)

\(x_{12} = |V(G)| - x_{11}\)

\(y_{11} = 16ni - 2n - i\)

\(y_{12} = |E(G)| - y_{11} - 2n\)

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