Mark Sequences and Sets in Directed Graphs

( Pirzada,Shariefuddin ) 한국수학교육학회 2011 수학교육 학술지 Vol.2011 No.-

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. One of the important theories is directed graphs, which represents the binary relations, and finds applications in many branches of sciences including computer sciences and networking. In directed graphs the concept of mark (score) attached to the vertex has been significantly investigated and used to study the structural properties. In this talk, we discuss various existence and constructive criterions for sequences of non-negative integers to be mark sequences and mark sets of various types of directed graphs. We also obtain algorithms for constructions of certain types of directed graphs.

Potential of PVA templated Silica Xerogels as Adsorbents for Rhodamine 6G

Pirzada, Tahira,Shah, Syed Sakhawat Korean Chemical Society 2011 대한화학회지 Vol.55 No.6

PVA/silica hybrid xerogels were synthesized by sonohydrolysis of a mixture of 2-way catalyzed TEOS and water solution of PVA. PVA was successfully removed from the xerogels through calcination and its removal was confirmed through TGA analysis of the calcined gel. Microstructure of the gels was studied through SEM, XRD and FTIR. Nitrogen sorption studies were conducted to find out surface area of different samples. It was found out that the samples having PVA removed through calcinations have higher surface area (411.64 $m^2$/g) than the samples (353.544 $m^2$/g) synthesized without any PVA. Adsorption properties of these xerogels synthesized by using different ratios of components were studied by taking Rhodamine G6 as a model adsorbate. The experiments were conducted at room temperature ($25^{\circ}C$). UV visible spectroscopy was used to measure the concentration of the dye before and after adsorption. The adsorption data of Rhodamine G6 on PVA modified silica is described by the Freundlich's adsorption model.

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.

ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS

Pirzada, Shariefuddin,Raja, Rameez Korean Mathematical Society 2016 대한수학회지 Vol.53 No.5

Let M be an R-module, where R is a commutative ring with identity 1 and let G(V,E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$. We show that M over R is finite if and only if the metric dimension of the graph $ann_f({\Gamma}(M_R))$ is finite. We further show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if M is a prime-multiplication-like R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if $$M{\sim_=}R$$. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M) is a prime ideal and Soc(M) = 0.

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 $n{\ge}k{\ge}2$.

Sargassum swartzii extracts ameliorate memory functions by neurochemical modulation in a rat model

Pirzada Jamal Ahmed Siddiqui,Adnan Khan,Nizam Uddin,Saima Khaliq,Munawwer Rasheed,Shazia Nawaz,Ahsana Dar,Muhammad Hanif 한국식품과학회 2017 Food Science and Biotechnology Vol.26 No.4

Recently, considerable attention has been paid to drug exploration from natural sources for treating memory loss, a major manifestation of various neurodegenerative diseases. Increasing evidences implicate brain serotonin metabolism in learning and memory, supporting the notion that targeting 5-HT (5-hydroxytryptamine) and its receptors would be beneficial in the treatment of cognitive disorders. In the present study, behavioral and neurochemical effects were examined following administration of Sargassum swartzii extracts in albino Wistar rats. Increase in spatial working memory and recognition memory was exhibited by the seaweed-treated rats as compared to controls. Plasma tryptophan, brain 5-HT, and 5-hydroxyindoleacetic acid levels were measured using HPLC–ECD, and a significant increase in brain 5-HT metabolism was observed in the seaweed-treated rats. The increase in memory functions following repeated administration of S. swartzii extracts is suggested to be due to the increased serotonergic neurotransmission in the brain of seaweed-treated rats.

ON SIGNLESS LAPLACIAN SPECTRUM OF THE ZERO DIVISOR GRAPHS OF THE RING ℤ_n

Pirzada, S.,Rather, Bilal A.,Shaban, Rezwan Ul,Merajuddin, Merajuddin The Kangwon-Kyungki Mathematical Society 2021 한국수학논문집 Vol.29 No.1

For a finite commutative ring R with identity 1 ≠ 0, the zero divisor graph ��(R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs ��(ℤn) for various values of n. Also, we find signless Laplacian spectrum of ��(ℤn) for n = pz, z ≥ 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph ��(ℤn) are signless Laplacian integral.

MARK SEQUENCES IN TRIPARTITE MULTIDIGRAPHS

Pirzada, S.,Samee, U. The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5

A tripartite $\gamma$-digraph is an orientation of a tripartite multigraph that is without loops and contains at most $\gamma$ 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 $\gamma$-scores), attached to the vertices of a tripartite $\gamma$-digraph.

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.