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On the Metric Dimension and Minimal Doubly Resolving Sets of Families of Graphs

Thesis Info

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Author

Sultan, Saba

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12931/1/Saba_Sultan_Maths_HSR_2018_GCU_lahore_03.08.2018.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726806985

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Let G = (V (G);E(G)) be a connected graph. The distance between two vertices u; v 2 V (G) is the length of shortest path between them and is denoted by d(u; v). A vertex x is said to resolve a pair of vertices u; v 2 V (G) if d(u; x) 6= d(v; x). For an ordered subset, B = fb1; b2; : : : ; bng of vertices of G, the n-tuple r(vjB) = (d(v; b1); d(v; b2); : : : ; d(v; bn)) is called representation of vertex v with respect to B or vector of metric coordinates of v with respect to B. The set B is called a resolving set of G if r(ujB) 6= r(vjB) for every pair of vertices u; v 2 V (G), i.e., the representation of each vertex with respect to B is unique. The resolving set with minimum cardinality is called metric basis of G. This minimum cardinality is called metric dimension and is denoted by _(G). Notice that the i-th coordinate in r(vjB) is 0 if and only if v = bi. Thus in order to show that B is a resolving set of G, it su_ces to verify that r(ujB) 6= r(vjB) for every pair of distinct vertices u; v 2 V (G) n B. Let G be a graph of order at least 2. Two vertices x; y 2 V (G) are said to doubly resolve the vertices u; v of G if d(u; x) ? d(u; y) 6= d(v; x) ? d(v; y): A subset D _ V (G) is called a doubly resolving set of G if every two distinct vertices of G are doubly resolved by some two vertices in D, i.e., all coordinates of the vector r(ujD)?r(vjD) can not be same for every pair of distinct vertices u; v 2 V (G). The minimal doubly resolving set problem is to _nd a doubly resolving set of G with the minimum cardinality. The cardinality of minimal doubly resolving set of G is denoted by(G). We have _(G) _(G) always. Therefore these sets can contribute in finding upper bounds on the metric dimension of graphs. In this thesis, we have investigated the minimal doubly resolving set problem for necklace graph, circulant graph, antiprism graph and M obius ladders. Also, in last part of thesis, the metric dimension problem has been investigated for kayak paddle graph and cycles with chord.
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68. Al-Qalam/The Pen

I/We begin by the Blessed Name of Allah

The Immensely Merciful to all, The Infinitely Compassionate to everyone.

68:01
a. Nun.
b. By the pen and
c. that which they write with it write.

68:02
a. O The Prophet!
b. You are not insane by the Grace of your Rabb - The Lord,

68:03
a. and, indeed, for you will be a reward never ending, never diminishing,

68:04
a. for, indeed, you are of an exalted status of moral excellence.

68:05
a. Soon you will see, and they – disbelievers – too will see,

68:06
a. which of you is insane.

68:07
a. Indeed, your Rabb - The Lord is Fully Aware of whoever strays off HIS Path,
b. as HE is also Fully Aware of those who are guided aright.

68:08
a. So do not yield to the pressures of those who persistently belie your Mission and the Divine Message.

68:09
a. They wish that you should compromise in your advocacy, so they too would compromise
on their attitude.

68:10
a. And do not yield to the pressure of any imprudent habitual oath-swearer,

68:11
a. to any slanderer, back biter,
b. going around spreading gossip to cause mischief among people,

68:12
a. who will hinder people from doing good, and
b. a defiant sinner, transgressor,

68:13
a. rude and moreover low-born,

68:14
a. who would so act merely because he possess wealth and children/sons and family influence.

68:15
a. Whenever OUR Messages...

Pengaruh Pengukuran Balanced Scorecard (BSC) Terhadap Kinerja Perguruan Tinggi (Studi Kasus Pada Badan Layanan Umum Universitas Islam Negeri Sultan Syarif Kasim Riau)

Penelitian ini bertujuan untuk melihat Pengaruh Pengukuran Balanced Scorecard (BSC) Terhadap Kinerja Perguruan Tinggi (Studi Kasus pada Badan Layanan Umum Universitas Islam Negeri Sultan Syarif Kasim Riau.  Penelitian ini menggunakan metode kuantitatif dan menggunakan data primer dan sekunder yang diolah dengan pertimbangan responden yang diperoleh dari kuesioner. Populasi penelitian ini adalah seluruh pegawai badan layanan umum tahun 2020 yang berjumlah 765 orang. Metode penetapan sampel yang digunakan adalah purposive sampling. Sampel dalam penelitian ini sebanyak 75 responden. Data diolah dengan menggunakan analisis regresi linear berganda dengan bantuan SPSS Versi 25. Hasil penelitian menunjukkan bahwa semua variabel penelitian  berpengaruh dan signifikan terhadap kinerja Badan Layanan Umum.

Evaluation of Pilot Scale Trickling Filter System for the Treatment of Wastewater

Attached growth wastewater treatment systems are considered to be effective in developing countries due to their low energy, operational and maintenance cost. A trickling filter system is a type of sewage treatment facility which is used to minimize the levels of carbonaceous compounds in terms of BOD5 and COD from wastewater in addition to pathogens and NH4-N level. The present research study was mainly focused on to treat domestic wastewater by pilot scale trickling filter systems. 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No significant reduction in most probable number (MPN) index of fecal coliforms was observed in the effluent in first 9 weeks of operation. However, a significant reduction in the MPN of fecal coliforms was observed, i.e. 80-90% in the last few weeks of treatment. In the 3rd phase, stone media setup-II TBF system was operated under different hydraulic retention time (HRT) i.e. 24, 48 and 72 hrs at a constant flow rate of 0.04 m3/day and it was perceived that stone media pilot scale TBF system showed high efficacy regarding removal of physicochemical and microbiological parameters under different HRTs. It was observed that by increasing HRT up to 48 and 72 hrs, the proficiency of setup-II TBF system increased significantly to reduce different parameters i.e. COD (70.9%), TDS (34%), EC (23.5%), SO4 (37%), PO4 (81.8%) and TN (66.6%). Furthermore, it was suggested that greater retention time and sand filtration play a key role in the pathogens removal and improvement of water quality. In the 4th phase, a new strategy was evaluated by using integrated media containing pebbles and gravels instead of stone to provide support for the growth of microbial slime layer. Furthermore, the integrated media TBF system was operated at three different flow rates (Q) i.e. 0.004 m3/day, 0.0072 m3/day and 0.01 m3/day and on each particular flow rate, effluent was operated under three different HRTs i.e. 48, 72 and 96 hrs in order to determine the efficiency of integrated media pilot scale TBF system treating domestic wastewater. It was observed that integrated bed material with air space had optimistic effect over TBF operation and the lowest and intermediate flow rates (0.004 m3/day and 0.0072 m3/day) showed promising results with respect to percent reduction of different physicochemical parameters i.e. COD (74.2-80.5%), TDS (60.3-69.5%), EC (62.8-68.6%) and PO4 (45.3-60.3%). A significant reduction in TN (59-63.3%) was observed at flow rates of 0.004 m3/day and 0.0072 m3/day. Moreover, it was observed that the efficiency of integrated media pilot scale TBF system in terms of pathogen removal (CFU/mL) increased significantly with continuous recirculation of wastewater for an extended period of time under different flow rates. From comparative assessment of stone media and integrated media pilot scale setup-II TBF system, it was evaluated that integrated media TBF system showed significant activity in the percent reduction of different physicochemical parameters as compared to stone media TBF system. Thus, overall results suggest that pilot-scale TBF has a great potential to be transferred to decentralized treatment system for handling sewage of small communities in developing countries, in order to produce effluent of good quality, which can be safely used for irrigation as well as ornamental purposes.