ہک واری تاں پھیرا پا
درشن اپنا آپ کرا
ناں تاں ساڈا سن لیندا اے
بھانویں یار ہے بے وفا
جیہڑی چنگیر تیری وچ روٹی
پڑھ بسم اللہ اوہنوں کھا
لگدا اس دی منگنی ہو گئی
ٹردی پئی چنی لٹکا
ملنا خوشی غمی تے چاہیدا
دنیا دا ہے رکھ رکھا
جو گدا بوہے تے آوے
خیر خوشی نال اوہنوں پا
ساری عمر گناہیں گزری
ہن تاں در توبہ دے آ
Citizen Satisfaction Index System (CSIS) was used to assess the delivery of support to education initiatives in the municipality of Banga, Aklan, Philippines. The samples were determined using multiple application of stratified random sampling approach. In accordance with the Philippine Statistical Authority's Data on Census Population and Housing for 2015, barangays having a bigger share of the population contributed more respondents to the 150 targeted participants. Following the inclusion criteria, the probability respondents were chosen using the Kish Grid. Pre-numbered questionnaires were distributed, with odd numbers targeting male responders and even numbers targeting females. The following criteria were used to evaluate the respondents' assessments: awareness, availment of the program/service, satisfaction, and need for action. Furthermore, interviews were conducted to better understand and investigate the respondents' thoughts, behavior, and perspectives. The reasons for their reaction were also obtained. The data was provided in percentage as well as frequency distributions. The study inferred that awareness on alternative learning system and/or special education program should be improved since it is the only program that attained low rating for awareness. Most of the residents were not able to avail education programs because they have no household member who attends school. Majority of the respondents who availed the services were satisfied. However, scholarship and other assistance programs to students may be enhanced to improve service delivery. It is highly recommended by the citizens to improve accessibility to scholarship programs and other forms of financial assistance to students.
The study of classical Ramsey numbers R(m, n) shows little progress in the last two decades. Only nine classical Ramsey numbers are known. This difficulty of finding the classical Ramsey numbers has inspired many people to study generalizations of classical Ramsey number. One of them is to determine Ramsey number R(G, H) for general graphs G and H (not necessarily complete). One of the most general results on graph Ramsey numbers is the establish- ment of a general lower bound by Chv ́atal and Harary [17] which is formulated as: R(G, H) ≥ (χ(H) − 1)(c(G) − 1) + 1, where G is a graph having no isolated vertices, χ(H) is the chromatic number of H and c(G) denotes the cardinality of large con- nected component of G. Recently, Surahmat and Tomescu [41] studied the Ramsey number of a combina- tion of path P n versus Jahangir graph J 2,m . They proved that R(P n , J 2,m ) = n+m−1 for m ≥ 3 and n ≥ (4m − 1)(m − 1) + 1. Furthermore, they determined that R(P 4 , J 2,2 ) = 6 and R(P n , J 2,2 ) = n + 1 for n ≥ 5. This dissertation studies the determination of Ramsey number for a combination of path P n and a wheel-like graph. What we mean by wheel-like graph, is a graph obtained from a wheel by a graph operation such as deletion or subdivision of the spoke edges. The classes of wheel-like graphs which we consider are Jahangir graph, generalized Jahangir graph and beaded wheel. First of all we evaluate the Ramsey number for path P n with respect to Jahangir graph J 2,m . We improve the result of Surahmat and Tomescu for m = 3, 4, 5 with n ≥ 2m + 1. Also, we determine the Ramsey number for disjoint union of k identical copies of path P n versus Jahangir graph J 2,m for m ≥ 2. Moreover, we determine the Ramsey number of path P n versus generalized Ja- hangir graph J s,m for different values of s, m and n. We also, evaluate the Ramsey number for combination of disjoint union of t identical copies of path versus general- ized Jahangir graph J s,m for even s ≥ 2 and m ≥ 3. At the end, we find the Ramsey number of path versus beaded wheel BW 2,m , i.e. R(P n , BW 2,m ) = 2n − 1 or 2n if m ≥ 3 is even or odd, respectively, provided n ≥ 2m 2 − 5m + 4.