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S. Pirzada,Ashay Dharwadker 한국산업응용수학회 2007 Journal of the Korean Society for Industrial and A Vol.11 No.4
Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. It is being actively used in fields as varied as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove fundamental results in other areas of pure mathematics. This paper, besides giving a general outlook of these facts, includes new graph theoretical proofs of Fermat's Little Theorem and the Nielson-Schreier Theorem. New applications to DNA sequencing (the SNP assembly problem) and computer network security (worm propagation) using minimum vertex covers in graphs are discussed. We also show how to apply edge coloring and matching in graphs for scheduling (the timetabling problem) and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. Finally, we revisit the classical problem of finding re-entrant knight's tours on a chessboard using Hamiltonian circuits in graphs.
Mark sequences in tripartite multidigraphs
S. Pirzada,U. Samee 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
A tripartite r-digraph is an orientation of a tripartite multigraph that is without loops and contains atmost r edges between any pair of vertices from distinct parts. In this paper, we obtain necessary and sufficient conditions for sequences of non-negative integers in non-decreasing order to be the sequences of numbers, called marks (or r-scores), attached to the vertices of a tripartite r-digraph. A tripartite r-digraph is an orientation of a tripartite multigraph that is without loops and contains atmost r edges between any pair of vertices from distinct parts. In this paper, we obtain necessary and sufficient conditions for sequences of non-negative integers in non-decreasing order to be the sequences of numbers, called marks (or r-scores), attached to the vertices of a tripartite r-digraph.
Score sequences in oriented graphs
S. PIRZADA,T. A. NAIKOO,N. A. SHAH 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
An oriented graph is a digraph with no symmetric pairs of directed arcs and without loops. The score of a vertex vi in an oriented graph D is avi ( or simply ai) = n − 1 + d+vi − d−vi , where d+vi and d−vi are the outdegree and indegree, respectively, of vi and n is the number of vertices in D. In this paper, we give a new proof of Avery’s theorem and obtain some stronger inequalities for scores in oriented graphs. We also characterize strongly transitive oriented graphs.
S. PIRZADA 한국산업응용수학회 2008 Journal of the Korean Society for Industrial and A Vol.12 No.1
The energy of a graph is the sum of the absolute values of its eigen values. We obtain some bounds for the energy of planar graphs in terms of its vertices, edges and faces.
Score sets in partite tournaments
S. Pirzada,T. A. Naikoo 한국전산응용수학회 2006 Journal of applied mathematics & informatics Vol.22 No.1-2
The set S of distinct scores (outdegrees) of the vertices of a kpartite tournament T(X1,X2, · · · ,Xk) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set of n non-negative integers is a score set of some k-partite tournament for every n k 2.
SOME MORE RESULTS IN CODING THEORY
S. Pirzada,Bilal Ahmad Bhat 한국산업응용수학회 2006 Journal of the Korean Society for Industrial and A Vol.10 No.2-1
In this paper;we have establish some generalized noiseless coding theorems by considering a non-additive measure of inaccuracy which is a generalization of non-additive measure of inaccuracy due to Gill et. al [7];under suitable conditions.
Signed degree sequences in signed 3-partite graphs
S. Pirzada,F. A. Dar 한국산업응용수학회 2007 Journal of the Korean Society for Industrial and A Vol.11 No.2
A signed 3-partite graph is a 3-partite graph in which each edge is assigned a positive or a negative sign. Let G(U, V, W) be a signed 3-partite graph with U = {u₁, u₂, …, up}, V = {v₁, v₂, …, vq} and W = {w₁, w₂, …, wr}. Then, signed degree of vi(vj and wk) is sdeg(ui) = di = di?-di?, 1≤i≤p(sdeg(vj))=ej=ej?-ej?, 1≤j≤q and sdeg(wk)=fk=fk?-fk?, 1≤k≤r) where di?(ej? and fk?) is the number of positive edges incident with ui(vj and wk) and di?(ej? and fk?) is the number of negative edges incident with ui(vj and wk). The sequences α = [d₁, d₂, …, dp], β = [e₁, e₂, …eq] and ν = [f₁, f₂, … fr] are called the signed degree sequences of G(U, V, W). In this paper, we characterize the signed degree sequences of signed 3-partite graphs.
Line graphs of unit graphs associated with the direct product of rings
S. Pirzada,Aaqib Altaf 강원경기수학회 2022 한국수학논문집 Vol.30 No.1
Let $R$ be a finite commutative ring with non zero identity. The unit graph of $R$ denoted by $G(R)$ is the graph obtained by setting all the elements of $R$ to be the vertices of a graph and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y \in U(R)$, where $U(R)$ denotes the set of units of $R$. In this paper, we find the commutative rings $R$ for which $G(R)$ is a line graph. Also, we find the rings for which the complements of unit graphs are line graphs.
SCORE SEQUENCES IN ORIENTED GRAPHS
Pirzada, S.,Naikoo, T.A.,Shah, N.A. 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
An oriented graph is a digraph with no symmetric pairs of directed arcs and without loops. The score of a vertex $v_i$ in an oriented graph D is $a_{v_i}\;(or\;simply\;a_i)=n-1+d_{v_i}^+-d_{v_i}^-,\;where\; d_{v_i}^+\;and\;d_{v_i}^-$ are the outdegree and indegree, respectively, of $v_i$ and n is the number of vertices in D. In this paper, we give a new proof of Avery's theorem and obtain some stronger inequalities for scores in oriented graphs. We also characterize strongly transitive oriented graphs.
SCORE SETS IN k-PARTITE TOURNAMENTS
Pirzada, S.,Naikoo, T.A. 한국전산응용수학회 2006 Journal of applied mathematics & informatics Vol.22 No.1
The set S of distinct scores (outdegrees) of the vertices of a k-partite tournament T($X_l,\;X_2, ..., X_k$) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set of n non-negative integers is a score set of some k-partite tournament for every $n{\ge}k{\ge}2$.