The Harmonic Index of Edge-Semitotal Graphs, Total Graphs and Related Sums

B. N. Onagh

For a connected graph $G$, there are several related graphs such as line graph $L(G)$, subdivision graph $S(G)$, vertex-semitotal graph $R(G)$, edge-semitotal graph $Q(G)$ and total graph $T(G)$ [I. Gutman, B. Furtula, �. K. Vukicevic and G. Popivoda, { On Zagreb indices and coindices}, MATCH Commun. Math. Comput. Chem. { 74} (2015), 5--16, W. Yan, B.-Y. Yang and Y.-N. Yeh, { The behavior of Wiener indices and polynomials of graphs under five graph decorations}, Appl. Math. Lett. { 20} (2007), 290--295]. Let $F$ be one of symbols $S$, $R$, $Q$ or $T$. The $F$-sum $G_1 ~ \!\! +_F ~\!\! G_2$ of two connected graphs $G_1$ and $G_2$ is a graph with vertex set $\left (V(G_1)\cup E(G_1) \right ) �V(G_2)$ in which two vertices $(u_1,v_1)$ and $(u_2,v_2)$ of $G_1 ~ \!\! +_F ~\!\! G_2$ are adjacent if and only if $\left [u_1=u_2 \in V(G_1)~\text{and} ~ v_1 v_2 \in E(G_2)\right ]$ or $\left [v_1=v_2 ~\text{and}~u_1u_2\in E(F(G)) \right ]$ [M.~Eliasi and B.~Taeri, { Four new sums of graphs and their Wiener indices}, Discrete Appl. Math. { 157} (2009), 794--803]. In this paper, we investigate the harmonic index of edge-semitotal graphs, total graphs and $F$-sum of graphs, where $F=Q$ or $T$.