Estimate of the Number of Edges in Subgraphs of a Johnson Graph

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained.

Analysis of statistic health care as indicators of efficiency service in Semen Padang Hospital through Barber Johnson graph

Background: The efficiency of service delivery is very important for hospitals. One measurement of service indicators that can be used is the Barber Johnson graph (GBJ). GBJ is needed to see and measure the level of service efficiency in hospitals. The indicators used are bed occupancy rate (BOR), bed turnover rate (BTR), turnover interval (TI), and length of stay (LOS). This graph can also be used to compare or view hospital developments at different times, and to increase the likelihood of changes in one variable by changing other variables. This research was conducted at Semen Padang Hospital (SPH), Padang, West Sumatera, Indonesia.Methods: The purpose of this study was to determine the statistical value of hospital and hospital service efficiency levels by using the Barber Johnson graphic. This research method is descriptive by direct observation of the medical record file of inpatients since the January to December 2017 period.Results: Statistical data obtained from SPH in 2018 showed the value of service days 30132, and the Number of beds 144 units. From the data processing results obtained a total bed occupancy rate 60.83%, bed turnover rate 6.86 times, turnover interval 2 days and average length of stay 3 days.Conclusions: Statistical data obtained from SPH in 2018 shows the value of BOR, TI is in an efficient, while BTR and LOS are inefficient.

On the Number of Matroids Compared to the Number of Sparse Paving Matroids

It has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that $\lim_{n\rightarrow\infty} s_n/m_n = 1$, where $m_n$ denotes the number of matroids on $n$ elements, and $s_n$ the number of sparse paving matroids. In this paper, we show that $$\lim_{n\rightarrow \infty}\frac{\log s_n}{\log m_n}=1.$$ We prove this by arguing that each matroid on $n$ elements has a faithful description consisting of a stable set of a Johnson graph together with a (by comparison) vanishing amount of other information, and using that stable sets in these Johnson graphs correspond one-to-one to sparse paving matroids on $n$ elements.As a consequence of our result, we find that for all $\beta > \displaystyle{\sqrt{\frac{\ln 2}{2}}} = 0.5887\cdots$, asymptotically almost all matroids on $n$ elements have rank in the range $n/2 \pm \beta\sqrt{n}$.

Cyclic Decomposition of $k$-Permutations and Eigenvalues of the Arrangement Graphs

The $(n,k)$-arrangement graph $A(n,k)$ is a graph with all the $k$-permutations of an $n$-element set as vertices where two $k$-permutations are adjacent if they agree in exactly $k-1$ positions. We introduce a cyclic decomposition for $k$-permutations and show that this gives rise to a very fine equitable partition of $A(n,k)$. This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of $A(n,k)$. Consequently, we determine the eigenvalues of $A(n,k)$ for small values of $k$. Finally, we show that any eigenvalue of the Johnson graph $J(n,k)$ is an eigenvalue of $A(n,k)$ and that $-k$ is the smallest eigenvalue of $A(n,k)$ with multiplicity ${\cal O}(n^k)$ for fixed $k$.

The Terwilliger Algebra of the Incidence Graphs of Johnson Geometry

In 2007, Levstein and Maldonado computed the Terwilliger algebra of the Johnson graph $J(n,m)$ when $3m\leq n$. It is well known that the halved graphs of the incidence graph $J(n,m,m+1)$ of Johnson geometry are Johnson graphs. In this paper, we determine the Terwilliger algebra of $J(n,m,m+1)$ when $3m\leq n$, give two bases of this algebra, and calculate its dimension.

On the connectedness and diameter of a geometric Johnson graph

Combinatorics International audience Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.