Arbind K. Lal

ZERO-SUM TWO-PERSON SEMI-MARKOV GAMES

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.

On Z4-Simplex Codes and Their Gray Images

In [7] Rains has shown that for any linear code C over Z4, dH, the minimum Hammimg distance of C and dL, the minimum Lee distance of C satisfy dH ≥ ⌈dL/2⌉. C is said to be of type α(β) if dH = ⌈dL/2⌉ (dH > ⌈dL/2⌉). In this paper we define Simplex codes of type α and β, namely, Skα and Skβ, respectively over Z4. Some fundamental properties like 2-dimension, Hamming and Lee weight distributions, weight hierarchy etc. are determined for these codes. It is shown that binary images of Skα and Skβ by the Gray map give rise to some interesting binary codes.

On Tanner Codes: Minimum Distance and Decoding

Abstract A bound on the minimum distance of Tanner codes / expander codes of Sipser and Spielman is obtained. Furthermore, a generalization of a decoding algorithm of Zémor to Tanner codes is presented. The algorithm can be implemented using the same complexity as that of Zémor and has a similar error-correction capability. Explicit families of Tanner codes are presented for which the decoding algorithm is applicable.

Laplacian Spectrum of Weakly Quasi-threshold Graphs

In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class of quasi-threshold graphs. We show that weakly quasi-threshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set. We also supply a quadratic time algorithm in the the size of the vertex set for recognizing such a graph. We completely determine the Laplacian spectrum of weakly quasi-threshold graphs. It turns out that weakly quasi-threshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs. A conjecture of Grone and Merris asserts that the spectrum of the Laplacian of any graph is majorized by the conjugate of the degree sequence of the graph. We show that the conjecture holds for cographs.

On the generalized Hamming weights of cyclic codes

We prove several results on the generalized Hamming weights (GHWs) of linear codes, particularly for cyclic codes. Based on these and previously known results, we give some efficient algorithms for computing GHW hierarchy of cyclic codes. We give complete weight hierarchy for each of the binary cyclic codes of odd lengths /spl les/31. A table of second and third GHWs of binary cyclic codes of odd lengths /spl les/57 is also presented. We have also computed the second GHW of all binary cyclic codes of length 63 and the third GHW of one code from each dimension.

Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices

In this paper, we consider the following problem. Over the class of all simple connected graphs of order n with k pendant vertices (n, k being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.

On generalized hamming weights and the covering radius of linear codes

We prove an upper bound on the covering radius of linear codes over IFq in terms of their generalized Hamming weights. We show that this bound is strengthened if we know that the codes satisfy the chain condition or a partial chain condition. We show that this bound improves all prior bounds. Necessary conditions for equality are also presented. Several applications of our bound are presented. We give tables of improved bounds on the covering radius of many cyclic codes using their generalized Hamming weights. We show that most cyclic codes of length ≤ 39 satisfy the chain condition or partial chain condition up to level 5. We use these results to derive tighter bounds on the covering radius of cyclic codes.

On algebraic connectivity of graphs with at most two points of articulation in each block

Let G be a connected graph and let L(G) be its Laplacian matrix. We show that given a graph G with a point of articulation u, and a spanning tree T, there is a way to give weights to the edges of G, so that u is the characteristic vertex and the monotonicity property holds on T. A restricted graph is a graph with a restriction that each block can have at most two points of articulation. We supply the structure of a restricted graph G whose algebraic connectivity is extremized among all restricted graphs with the same blocks as those of G. Further results are supplied when each block of G is complete. A path bundle is a graph that consists of internally vertex disjoint paths of the same length with common end vertices. Results pertaining to extremizing the algebraic connectivity of restricted graphs whose blocks are path bundles are supplied. As an application, a comparison of the algebraic connectivities of the sunflower graphs is provided.

The distance matrix of a bidirected tree

A bidirected tree is a tree in which each edge is replaced by two arcs in either direction. Formulas are obtained for the determinant and the inverse of a bidirected tree, generalizing well-known formulas in the literature.

Path-positive graphs

For any positive integer k let P, denote the characteristic polynomial of the adjacency matrix of the path on k vertices. We obtain certain results regarding t’, evaluated at A(C), the adjacency matrix of any given graph G. For some special graphs G, we describe the matrix P,(A(G)) completely. We call a graph G path-positive if I’,(A(G)) is a nonnegative matrix for all positive integers k. It is shown that if G is a connected graph with a vertex of degree at least four, then G is path-positive. We conjecture that all connected graphs are path-positive apart from a few exceptional cases described in the paper.