Isometric-path numbers of block graphs

Isometric-path numbers of block graphs

An isometric path between two vertices in a graph G is a shortest path joining them. The isometric-path number of G, denoted by ip(G), is the minimum number of isometric paths required to cover all vertices of G. In this paper, we determine exact values of isometric-path numbers of block graphs. We also give a linear-time algorithm for finding the corresponding paths.

Incidence dominating numbers of graphs

In this paper, the concept of incidence domination number of graphs  is introduced and the incidence dominating set and  the incidence domination number  of some particular graphs such as  paths, cycles, wheels, complete graphs and stars are studied.

Energy of Graphs, Matroids and Fibonacci Numbers

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.

Fractional isometric path number

Corrigendum to "The path-partition problem in block graphs"

Recently, Wong [1] pointed out that Yan and Chang’s [2] linear-time algorithm for the path-partition problem for block graphs is not correct, by giving the following example. Suppose G is the graph consisting of a vertex w and a set of triangles {xi, yi, zi} such that each xi is adjacent to w for 1 i k, where k 3. Then p(G) = k − 1, but Yan and Chang’s algorithm gives p(G)= 1. He also traced th...