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Dhananjay P. Mehendale

Publications and source records attributed to Dhananjay P. Mehendale.

At least 19 recordsLinked to original sources
arXiv2025

Hamiltonian Graphs and the Traveling Salesman Problem

A new characterization of Hamiltonian graphs using f-cutset matrix is proposed. Based on this new characterization, a new exact polynomial time algorithm for the traveling salesman problem (TSP) is developed. We then define the so-called ordered weighted adjacency list for given weighted complete graph and proceed to the paper's main result, namely, the exact algorithm based on the utilization of the ordered weighted adjacency list and the simple properties that any path or circuit must satisfy. This algorithm performs checking of sub-lists, containing (p-1) entries (edge pairs) for paths and p entries (edge pairs) for circuits, chosen from ordered adjacency list in a well defined sequence to determine exactly the shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph of p vertices. The procedure has intrinsic advantage of landing on the desired solution in quickest possible time and even in worst case in polynomial time. A new characterization of the shortest Hamiltonian tour for a weighted complete graph satisfying triangle inequality (i.e. for tours passing through every city on a realistic map of cities where cities can be taken as points on a Euclidean plane) is also proposed. Finally, we propose a classical algorithm for unstructured search, three new quantum algorithms for unstructured search, which exponentially speed up the searching ability in the unstructured database, and one quantum algorithm for solving a K-SAT problem and indicate its effect on traveling salesman problem and other NP-complete problems.

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arXiv2024

On Gracefully Labeling Trees

In this paper, we propose an algorithm to generate all possible graceful graphs (including trees) containing n vertices as lattice paths in a certain triangular lattice defined below. This lattice that corresponds to graphs containing n vertices is called an n-lattice and is made up of certain rows of vertex pairs (i, j). Each row of this n-lattice is made up of those vertex-pairs, say (i, j), for which the difference|i - j| is the same for every vertex-pair belonging to that row, and where i, j belongs to set {1, 2, ..., n}. The first row of this n-lattice contains (n - 1) vertex pairs, (i, i + 1), i = 1, 2, ..., (n - 1). The second row of this n-lattice contains (n - 2) vertex pairs, (i, i + 2), i = 1, 2, ..., (n - 2). In this way, one goes down to the last row of this lattice which contains only one vertex-pair, (1, n). A lattice path is one made up of (n - 1) vertex pairs such that every row of the triangular lattice contributes exactly one vertex pair to this lattice path. We obtain all possible lattice paths without omission or repetition by generating them in a systematic way, in a well-defined lexicographic order. The collection of all such lattice paths forms all possible graceful graphs. We will note various observations related to these lattice paths. For example, the lattice paths appear in symmetric pairs, i.e. for each lattice path there exists a corresponding unique lattice path which is the mirror image of this lattice path taken in the line of symmetry passing vertically and centrally through the lattice, each lattice path and its corresponding mirror image represent isomorphic graceful graphs. The main result of this paper is the affirmative settlement of the well-known graceful tree conjecture.

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arXiv2023

On Entanglement and Separability

We present a necessary and sufficient condition to determine the entanglement status of an arbitrary N-qubit quantum state (may be pure or mixed) represented by the density matrix, (Rho)N. We develop a new approach and a new criterion for the problem of deciding entanglement status. Further, we develop as an important application of entanglement a new quantum protocol for superluminal communication of classical information in terms of a desired ordered sequence of classical bits. We then show that this new quantum protocol for superluminal communication of classical information can be used in the quantum teleportation protocol [12] for achieving superluminal quantum teleportation.

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arXiv2019

A New Algorithm for Linear Programming

We propose a new polynomial-time algorithm for linear programming. We further extend the ideas used in this new linear programming algorithm for nonlinear programming problems. The new algorithm is based on the idea of treating the objective function as a parameter. We form a matrix of coefficients, made-up of the coefficients of the variables defined in the problem itself, and the coefficients of variables defined newly, for converting inequalities into equations, namely, slack variables if it is the maximization problem, or, surplus variables if it is the minimization problem. The system of equations we use consist of the objective equation and equations obtained from inequalities defining constraint imposed by the problem. We obtain reduced-row-echelon-form, R, for this matrix containing only one unknown, namely, the objective function itself as an unknown parameter, d, say. This matrix in the reduced-row-echelon-form contains columns (column vectors) corresponding to basic variables and non-basic variables. If all the entries in the columns corresponding to non-basic variables in R are already nonnegative then we will see that we have almost reached to the solution and nothing much is left to be done. If there are columns corresponding to non-basic variables which contain some negative entries then we will require to apply suitable row transformations, at most $m$ in number if there are m rows in R, as we will see below, to make all the entries in the columns corresponding to non-basic variables nonnegative. We then proceed to show that the method developed above for linear programming naturally extends to nonlinear programming problems. For nonlinear programming problems, we use the technique of Grobner bases, since Grobner basis is an equivalent of reduced row echelon form for a system of nonlinear equations.

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arXiv2016

On the Jacobian Question

The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic approach is shown to be extendable to higher dimensions by proceeding on exactly identical lines. As per the important result due to Bass, Connell, and Wright [3] it is enough to show the validity of the Jacobian conjecture for cubic polynomials of special type (BCW form) in two, three, and n variables. Firstly, the method of proof for the cases of two and three variables is discussed at length. It is then indicated that the extension to the several variables case follows automatically by just following the same steps and there is no hindrance as there is essentially no change in the basic situation and the same line of thought used for the case of two and three variables remains applicable. Thus, we show that the problem can be solved completely using the important reduction of the problem to the case of special cubic degree polynomials [3]. We have shown how to obtain inverse polynomials. We fully obtain them for two variables case and almost obtain them for three variables case.

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arXiv2015

On Isomorphism of Graphs and the k-clique Problem

In this paper we develop three characterizations for isomorphism of graphs. The first characterization is obtained by associating certain bitableaux with the graphs. We order these bitableaux by suitably defined lexicographic order and denote the bitableau that is least in this order as the standard representation for the associated graph. The standard representation characterizes graphs uniquely. The second characterization is obtained in terms of associated rooted, unordered, pseudo trees. We show that the isomorphism of two given graphs is implied by the isomorphism of their associated pseudo trees. The third characterization is obtained in terms of ordered adjacency lists to be associated with two given labeled graphs. We show the two given labeled graphs are isomorphic if and only if their associated ordered adjacency lists can be made identical by the action of suitable transpositions on any one of these lists. We discuss in brief the complexity of these characterizations described in this paper for deciding isomorphism of graphs. Finally, we discuss the k-clique problem in the light of these characterizations towards the end of the paper.

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arXiv2015

On Problems Related to Primes: Some Ideas

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their infinitude; establishment of a kind of similarity between natural numbers and numbers that appear in an arithmetic progression, similar formulae for primes and the so called generalized twin primes in an arithmetic progression and their infinitude; generalization of Bertrand postulate and a Bertrand like postulate for twin primes; some elementary implications of a simple primality test, the use of Chinese remainder theorem in a possible proof of the Goldbach conjecture; Schinzel Sierpinski conjecture; and the Mersenne primes and composites, Fermat primes and their infinitude. Lastly we define other twin primes and provide a simple argument in support of their infinitude.

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arXiv2015

Finite Projective Planes

We propose graph theoretic equivalents for existence of a finite projective plane. We then develop a new approach and see that the problem of existence of a finite projective plane of order n is linked up with a subset of sharply 2 transitive permutations. If n is prime power then it is well known that there exists a finite field and existence of this field implies existence of MOLS which further implies existence of fpp. We show that by assuming the existence of MOLS the existence of a group made up of sharply 2 transitive permutations can be implied through transforming the given MOLS to suitable form. From a known results it then follows that when such group exists the order n has to be a prime power. We then see the relation between MOLS and determinantal monomials and between MOLS and a cyclic group that permutes the rows of MOLS. Finally, we conclude the paper with some important remarks.

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arXiv2011

The Reconstruction of Graphs

In this paper we discuss reconstruction problems for graphs. We develop some new ideas like isomorphic extension of isomorphic graphs, partitioning of vertex sets into sets of equivalent points, subdeck property, etc. and develop an approach to deal with reconstruction problem. We then discuss complete sets of invariants for graphs and reconstruction conjecture. We then begin with development of few equivalent formulations of reconstruction conjecture. In the last section we briefly elaborate the formulation due to Harary its exact demand and finally proceed to give a different proof of reconstruction conjecture using reconstructibility of graph from its spanning trees and reconstructibility of tree from its pendant point deleted deck of subtrees. This last proof can be used to develop a systematic procedure to reconstruct unique graph from its deck.

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arXiv2011

On Path Decomposition Conjecture of Tibor Gallai

We settle the Path Decomposition Conjecture (P.D.C.) due to Tibor Gallai for minimally connected graphs, i.e. trees. We use this validity for trees and settle the P. D. C. using induction on the number of edges for all connected graphs. We then obtain a new bound for the number of paths in a path cover in terms of the number of edges using idea of associating a tree with a connected graph. We then make use of a spanning tree in the given connected graph and its associated basic path cover to settle the conjecture of Tibor Gallai in an alternative way. Finally, we show the existence of Hamiltonian path cover satisfying Gallai bound for complete graphs of even order and discuss some of its possible ramifications.

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arXiv2010

A New Algorithm for Multicommodity Flow

We propose a new algorithm to obtain max flow for the multicommodity flow. This algorithm utilizes the max-flow min-cut theorem and the well known labeling algorithm due to Ford and Fulkerson [1]. We proceed as follows: We select one source/sink pair among the n distinguished source/sink pairs at a time and treat the given multicommodity network as a single commodity network for such chosen source/sink pair. Then applying standard labeling algorithm, separately for each sink/source pair, the feasible flow which is max flow and the corresponding minimum cut corresponding to each source/sink pair is obtained. A record is made of these cuts and the paths flowing through the edges of these cuts. This record is then utilized to develop our algorithm to obtain max flow for multicommodity flow. In this paper we have pinpointed the difficulty behind not getting a max flow min cut type theorem for multicommodity flow and found out a remedy.

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arXiv2008

On Hamilton Decompositions

P. J. Kelly conjectured in 1968 that every diregular tournament on (2n+1) points can be decomposed in directed Hamilton circuits [1]. We define so called leading diregular tournament on (2n+1) points and show that it can be decomposed in directed Hamilton circuits when (2n+1) is a prime number. When (2n+1) is not a prime number this method does not work and we will need to devise some another method. We also propose a general method to find Hamilton decomposition of certain tournament for all sizes.

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arXiv2008

On Caccetta-Haggkvist Conjecture

We show that we cannot avoid the existence of at least one directed circuit of length less than or equal to (n/r) in a digraph on n vertices with out-degree greater than or equal to r. This is well-known Caccetta-Haggkvist problem.

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arXiv2008

Ising Problem on Simple Cubic Lattice

Simple cubic lattice (SC lattice) can be viewed as plane triangular lattice (PT lattice) by viewing it along its principle diagonal lines. By viewing thus we establish the exact one-to-one correspondence between the closed graphs on SC lattice and the corresponding closed graphs on PT lattice. We thus see that the propagator for PT lattice (with suitable modifications) can be used to solve, at least in principle, the 3D Ising problem for SC lattice in the absence of external magnetic field. A new method is then proposed to generate high temperature expansion for the partition function. This method is applicable to 2D as well as 3D lattices. This method does not require explicit counting of closed graphs and this counting is achieved in an indirect way and thus exact series expansion can be achieved up to any sufficiently large order.

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arXiv2007

On Hadwiger Conjecture

We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then define correct contraction procedure to be executed to get largest possible complete graph from given connected graph. We finally give a characterization for k-chromatic graphs and use it to settle Hadwigers conjecture.

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arXiv2007

On the Theory of Colorful Graphs

The theory of colorful graphs can be developed by working in Galois field modulo (p), p > 2 and a prime number. The paper proposes a program of possible conversion of graph theory into a pleasant colorful appearance. We propose to paint the usual black (indicating presence of an edge) and white (indicating absence of an edge) edges of graphs using multitude of colors and study their properties. All colorful graphs considered here are simple, i.e. not having any multiple edges or self-loops. This paper is an invitation to the program of generalizing usual graph theory in this direction.

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arXiv2006

On Ramsey Numbers

In this paper we define new numbers called the Neo-Ramsay numbers. We show that these numbers are in fact equal to the Ramsay numbers. Neo-Ramsey numbers are easy to compute and for finding them it is not necessary to check all possible graphs but enough to check only special kind of graphs having a well-defined adjacency pattern.

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