Iranian Journal of Mathematical Chemistry
Volume:11 Issue: 4, Autumn 2020

The modified first Zagreb connection index $ZC_{1}^{*}$ for a graph $G$ is defined as $ZC_{1}^{*}(G)= sum_{vin V(G)}d_{v}tau_{v},$, where $d_{v}$ is degree of the vertex $v$ and $tau _{v}$ is the connection number of $v$ (that is, the number of vertices having distance 2 from $v$). By an $n$-vertex graph, we mean a graph of order $n$. A branching vertex of a graph is a vertex with degree greater than $2$. In this paper, the graphs with maximum and minimum $ZC_{1}^{*}$ values are characterized from the class of all $n$-vertex trees with a fixed number of branching vertices.
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The leap eccentric connectivity index of $G$ is defined as $$Lxi^{C}(G)=sum_{vin V(G)}d_{2}(v|G)e(v|G)$$ where $d_{2}(v|G) $ be the second degree of the vertex $v$ and $e(v|G)$ be the eccentricity of the vertex $v$ in $G$. In this paper, we give some properties of the leap eccentric connectivity index of the graph $G$.
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The liquid crystal (LC) 4-4′-disubstituted biphenyls (HnCBP) of the general line formula HO-(CnH2n+1)-O-C6H4-C6H4-CN (n=1-12) shows the odd-even effect under the applied electric field. The odd-even effects are observed in the HOMO-LUMO gap, birefringence, order parameter, and dipole moment. The odd carbon atom number of alkyl chain shows HOMO-LUMO gap, birefringence and order parameter in the upward direction and even carbon atom number of alkyl chain shows in the downward direction; however the dipole moment exhibits a shift of even carbon number of alkyl chain in the upward direction and odd carbon number of alkyl chain in the downward direction.

The Gutman index, Schultz index, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index are four well-studied topological indices, which are useful tools in QSPR and QSAR investigations. Spiro compounds are an important class of cycloalkanes in organic chemistry. In this paper, we determine the expected values of these indices in the random spiro chains, and the extremal values among all spiro chains with n hexagons.

The Merrifield-Simmons index of a graph G is the number of independent sets in G. In this paper, we give exact formulae for the expected value of the Merrifield-Simmons index of random phenylene chains by means of auxiliary graphs.