In this paper, we have characterized the commutative rings with unity for which Line signed graph of a signed unit graph is balanced and consistent. To do this, we first establish some sufficient conditions for balance and consistency of Line signed graph of signed unit graphs.

Expressions for the Zagreb indices and coindices of the total graph, semitotal point graph and of semi-total Line graph of subdivision graphs in terms of the parameters of the parent graph are obtained, thus generalizing earlier existing results.

The concept of super Line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super Line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The Line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we  nd the Line completion number of grid graph Pn  Pm for various cases of n and m.

The Bertz indices, derived by counting the number of connecting edges of Line graphs of a molecule were used in deriving the QSPR models for the physicochemical properties of alkanes. The inability of these indices to identify the hetero centre in a chemical compound restricted their applications to hydrocarbons only. In the present work, a novel molecular descriptor has been derived from the weighted Line graph of the molecular structure and applied in correlating the physicochemical properties of alkane isomers with these descriptors. A weight is tagged at the vertex of the Line graph, which consequently modifies the weight of the edge. These descriptors were found to classify the alkane isomers and served well in deriving the QSPR models for various physicochemical properties. The mathematical calculations include the quantitative treatment on the role of substituents (alkyl) in governing the properties under study of the alkane isomers. Further, the use of weighted Line graph in the enumeration of the topological index opens up a new vista on application to heteroatomic systems.

The metric dimension of a graph is a fundamental parameter that measures the minimum number of vertices to identify all other vertices in the graph uniquely‎. ‎In the context of chemical structures‎, ‎where graphs represent molecular entities‎, ‎the metric dimension becomes a crucial metric for understanding molecular behavior and interactions‎. ‎A subset $T = \{ t_1,t_2,\ldots‎, ‎t_k \}$ of nodes of a connected network $G$ is referred to as a revolving set‎, ‎if for any pair of nodes‎, ‎$ l,m \in V(G)$ there exists a node $t \in T$‎, ‎such that its distances from $l$ and $m$ are different‎. ‎The smallest {cardinality} of $T$ is referred to as the metric dimension of $G$‎, ‎and the nodes in $T$ constitute a metric basis of $G$‎. ‎In this work‎, ‎we calculate the Line graph's metric dimension for some chemical structures such as hexagon-square chains‎, ‎Linear phenylene structures‎, ‎and Linear heptagonal structures‎.

Let $G=(V,E)$, $V=\left\{ v_{1},v_{2},\ldots ,v_{n}\right\}$, be a simple graph of order $n$ and size $m$.Denote by $\Delta = d_1\ge d_2 \ge \cdots \ge d_n= \delta$, $d_i=d(v_i)$, and $\Delta_e=d(e_1)\ge d(e_2)\ge \cdots \ge d(e_m)=\delta_e$, sequences of vertex and edge degrees, respectively. The first reformulated Zagreb index (coindex) is defined as $\displaystyle EM_1(G)=\sum_{i=1}^m d(e_i)^2 = \sum_{e_i\sim e_j}(d(e_i)+d(e_j))$ $\Big(\displaystyle \overLine{EM}_1(G) = \sum_{e_i\nsim e_j}(d(e_i)+d(e_j))\Big)$.We consider relationship between reformulated Zagreb indices/coindices and determine their bounds in terms of some basic graph parameters.

Let R be a commutative ring with identity. We use j (R) to denote the comaximal ideal graph. The vertices of j (R) are proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I and J are adjacent if and only if I+J=R. In this paper we show some properties of this graph together with planarity of Line graph associated to j (R).