Complexity to Find Wiener Index of Some Graphs

The Wiener index is one of the oldest graph parameter which is used to study molecular-graph-based structure. This parameter was first proposed by Harold Wiener in 1947 to determining the boiling point of paraffin. The Wiener index of a molecular graph measures the compactness of the underlying molecule. This parameter is wide studied area for molecular chemistry. It is used to study the physio-chemical properties of the underlying organic compounds. The Wiener index of a connected graph is denoted by W(G) and is defined as, that is W(G) is the sum of distances between all pairs (ordered) of vertices of G. In this paper, we give the algorithmic idea to find the Wiener index of some graphs, like cactus graphs and intersection graphs, viz. interval, circular-arc, permutation, trapezoid graphs.Comment: 6 page

Efficient algorithm for the vertex connectivity of trapezoid graphs

The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. These graphs and their generalizations were applied in various fields, including modeling channel routing problems in VLSI design and identifying the optimal chain of non-overlapping fragments in bioinformatics. Using modified binary indexed tree data structure, we design an algorithm for calculating the vertex connectivity of trapezoid graph $G$ with time complexity $O (n \log n)$, where $n$ is the number of trapezoids. Furthermore, we establish sufficient and necessary condition for a trapezoid graph $G$ to be bipartite and characterize trees that can be represented as trapezoid graphs.Comment: 12 pages, 2 figure

L(2,1)-labelling of Circular-arc Graph

An L(2,1)-labelling of a graph $G=(V, E)$ is $\lambda_{2,1}(G)$ a function $f$ from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. The L(2,1)-labelling number denoted by $\lambda_{2,1}(G)$ of $G$ is the minimum range of labels over all such labelling. In this article, it is shown that, for a circular-arc graph $G$, the upper bound of $\lambda_{2,1}(G)$ is $\Delta+3\omega$, where $\Delta$ and $\omega$ represents the maximum degree of the vertices and size of maximum clique respectively.Comment: 12 page