Distance-regular graph with large a1 or c2

In this paper, we study distance-regular graphs $\Gamma$ that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of $\Gamma$. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.Comment: We submited this manuscript to JCT

Shilla distance-regular graphs

A Shilla distance-regular graph G (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph G, and for G we define b=b(G):=k/a3. In this paper we will show that there are finitely many Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.Comment: 14 page

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.Comment: 15 pages, this is submitted to Linear Algebra and Applications

The effect of perception of a successful retired life and attitude towards retirement on retirement plans among nursing managers in clinics

Background: The concept of successful aging is comprehensive, and the meaning the concept is vague, thus the consensus thereof is yet to be clarified. And the quality of elderly-age life is dependent on provisions prepared before reaching elderly age. In this line, the purpose of this study was to identify the effect of perception of successful retired life and the attitude toward retirement regarding nursing managers’ retirement plans to assure a stable retired life for nurses. Methods and tools: Employing a structured questionnaire used for a previously conducted study, data were collected in December 2018 to February 2019 from nursing managers working at four high-class general hospitals. The appropriate sample size was calculated using the G*Power 3.1.9.2 program. Questionnaires were distributed to 157 subjects; a total of 141 copies were examined for the analysis of perceptions of a successful retired life and attitudes towards retirement on retirement plans, by conducting a t-test, variance analysis, Pearson’s correlation analysis, and multiple regression analysis. Results: Academic background and marital status showed insignificant differences in terms of the degree of perception of successful retirement, attitudes toward retirement, and provisions for retirement. The nursing managers age 50s-60s have positive attitudes toward respective times of retiree life comprising overall provisions including economic and physical as well than nursing managers of age 40 and under 44 , whereas the nursing managers of careers in nursing older than age 30 years and above prefer to focus on economic provisions. .Nursing managers with careers in nursing of more than 10 years exhibited more positive attitudes toward retirement than nursing managers with careers of less than five years. The perception of a successful retired life (β =.171, p&lt;.05) and attitude towards retirement (β =.265, p&lt;.01) exhibited a positive effect on nursing managers’ retirement plans. Conclusions: The results of this study are expected to be used as basic data for raising awareness among nurses of the importance of establishing stable retirement plans. Older nursing managers in the clinics, who exhibited better provisions against respective retirement in accordance with extended careers correspondingly, were attributed to their positions in the stage of each life ahead of retirement wherein they focused on preparing provisions against times after retirement with their children who had mostly graduated from mandatory courses of education. [Ethiop. J. Health Dev. 2020;34(Special issue-3):03-09] Key words: Attitude, nursing manager, perception, preparation, retiremen

Geometric aspects of 2-walk-regular graphs

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show