Ashok T. Amin

Parity dimension for graphs

Abstract For a graph G = ( V , E ), by N = A + I , we denote the closed neighborhood matrix of G where A and I are the adjacency matrix of G and identity matrix, respectively. The parity dimension of G , denoted PD( G ), is the dimension of the null space of N over the field Z 2 . We investigate parity dimension for trees, graphs and random graphs.

On the nonexistence of uniformly optimal graphs for pair-connected reliability

We consider probabilistic graphs G = (V, E) in which each edge xy ∈ E fails independently with probability q. The reliability measure studied is pair-connectivity, the expected number of pairs of connected vertices. We examine how the coefficients of the pair-connected reliability polynomial are determined by the subgraph structure of G, and we use these results to show that in most cases there does not exist a uniformly optimal n-vertex, m-edge graph.

Parity Dimension for Graphs - A Linear Algebraic Approach

For a graph G with closed neighborhood matrix N , the parity dimension of G , denoted PD( G ), is the dimension of the null space of N over the field

On uniformly optimally reliable graphs for pair‐connected reliability with vertex failures

{\cal Z}_2

The optimal unicyclic graphs for pair-connected reliability

. Equivalently, the number of vertex sets S in G with the property that S dominates each vertex an even number of times is 2 k for some value of k , and PD( G ) = k . Using primarily linear algebraic techniques, we investigate the parity dimension of graphs.

Algorithms for optimal orientations of a unicyclic graph

Let G be a probabilistic (n,m) graph in which each vertex exists independently with fixed probability p, 0 < p < 1. Pair-connected reliability of G, denoted PCv(G;p), is the expected number of connected pairs of vertices in G. An (n,m) graph G is uniformly optimally reliable if PCv(G;p) ≧ PCv(H;p) for all p, 0 < p < 1, over all (n,m) graphs H. It is shown that there does not exist a uniformly optimally reliable (n,m) graph whenever n ≦ m < ∼2n2/9. However, such graphs do exist for some other values of m. In particular, it is established that every complete k-partite pseudoregular graph on n vertices, 2 ≦ k < n, is uniformly optimally reliable.

Pair-Connected Reliability of a Tree and its Distance Degree Sequences

Abstract We consider the standard network reliability model in which each edge of a graph fails, independently of all others, with probability q = 1 − p (0 ≤ p ≤ 1). The pair-connected reliability of the graph is the expected number of pairs of vertices that remain connected after the edge failures. The optimal graphs for pair-connected reliability in the class of unicyclic graphs (connected ( n , n ) graphs) are completely characterized. The limiting behavior of the intervals of optimality are studied as n → ∞.

Neighborhood Domination with Parity Restriction in Graphs

Let (<i>G</i>, <i>R</i>) denote the directed graph obtained from undirected graph <i>G</i> by an acyclic orientation <i>R</i> so that (<i>G</i>, <i>R</i>) contains no directed cycle. We consider acyclic orientations <i>R</i> of a unicyclic graph <i>G</i> which maximizes/minimizes the number of ordered pairs of non-adjacent vertices with directed paths in (<i>G</i>, <i>R</i>). These orientations are referred as optimal orientations. We present algorithms to determine optimal orientations of a unicyclic graph.