Improved lower bounds on the extrema of eigenvalues of graphs

In this note, we improve the lower bounds for the maximum size of the $k$th largest eigenvalue of the adjacency matrix of a graph for several values of $k$. In particular, we show that closed blowups of the icosahedral graph improve the lower bound for the maximum size of the fourth largest eigenvalue of a graph, answering a question of Nikiforov.Comment: Minor revision

Short proofs of three results about intersecting systems

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$, which improves upon a recent result of O'Neill and Verstra\"{e}te. Our proof also extends to $d$-wise, $t$-intersecting families, and from this result we obtain a version of the Erd\H{o}s-Ko-Rado theorem for $d$-wise, $t$-intersecting families. The second result partially proves a conjecture of Frankl and Tokushige about $k$-uniform families with restricted pairwise intersection sizes. The third result concerns graph intersections. Answering a question of Ellis, we construct $K_{s, t}$-intersecting families of graphs which have size larger than the Erd\H{o}s-Ko-Rado-type construction whenever $t$ is sufficiently large in terms of $s$.Comment: 12 pages; we added a new result, Theorem 1

Maximum spread of K2,tK_{2,t}-minor-free graphs

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there is an integer $\xi_t$ such that the maximum spread of an $n$-vertex $K_{2,t}$-minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor$ copies of $K_t$ and $n-1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. The extremal graph is unique, except when $t\equiv 4 \mod 12$ and $\frac{2n+ \xi_t} {3t}$ is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor-1$ copies of $K_t$ and $n-1-t(\lfloor \frac{2n+\xi_t}{3t}\rfloor-1)$ isolated vertices. Furthermore, we give an explicit formula for $\xi_t$.Comment: 15 pages. arXiv admin note: text overlap with arXiv:2209.1377

Derangements in a Ferrers Board

The classic derangement question of counting the number of derangements for n objects from some initial permutation of the objects was first considered by de Montfort in 1708. A particular recasting of a permutation allows us to place any permutation onto an n x n board, from which certain properties of derangements may be understood. This research extends the classic derangement question to the more general Ferrers board, which is an n x n board with a missing section in the lower-right corner. Various properties of the derangement numbers for these more general boards are stated and proven in the course of this work

The domination number of the graph defined by two levels of the n-cube, II.

Consider all k-element subsets and ℓ-element subsets (k>ℓ) of an n-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding ℓ-element set is a subset of the corresponding k-element set. Let Gk,ℓ denote this graph. The domination number of Gk,1 was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value (n tending to infinity) of the domination number of Gk,2. Here we prove the conjecture, determining the asymptotic value of the domination number [Formula presented]. © 2020 The Author

Investigation of a Markov Chain on Ferrers Boards

This thesis is an investigation of some of the basic combinatorial, algebraic and probabilistic properties of a Markov chain on Ferrers Boards (i.e., a Markov chain whose states are permutations on a given Ferrers Board). This is an extension of extensive work done over the last fifty years to understand the properties of a Markov chain known as the Tsetlin library. We will review the extensive literature surrounding the Tsetlin library, which also allows for the problem to be contextualized as a particularly nice model of a procedure for searching a database of files. Some of the specific questions we will explore include the transitivity of the Tsetlin library (in fact, we will prove that the extended library is transitive and at most n steps are needed to reach any state from an arbitrarily chosen state); the Tsetlin library’s relation to permutation inversions and some other combinatorial statistics; and finally the computation of the Tsetlin library’s stationary distribution and eigenvalues in some easy cases. Although our analysis of the combinatorial aspects of the extended Tsetlin library is complete, we have been unable to fully describe the probabilistic aspects of the Tsetlin library. We are able to describe the stationary distribution for specific easy cases, but further analysis for more complicated cases has proven difficult. Computations have been done using the mathematical software Maple to determine if any patterns may be discerned from specific examples of the more complicated cases. However, the data indicates that the actual stationary distribution differs from our conjectured formula for the stationary distribution, which gives a need for further analysis in future work. We have also not been able to describe the eigenvalues or convergence to stationary for even the simplest Ferrers boards, but we do have various computations which we hope will be the basis for future exploration of these topics

Winter Habitat Use and Survival of Female Ring-necked Pheasants (\u3ci\u3ePhasianus colchicus\u3c/i\u3e) in Southeastern North Dakota

From 1992 to 1995 we used radiotelemetry to monitor winter habitat selection and survival of female ring-necked pheasants (Phasianus colchicus) in southeastern North Dakota. We captured 100 birds at nine sites in six study blocks centered on cattail-dominated (Typha spp.) semi-permanent wetlands. Pheasants showed nonrandom habitat use at two hierarchical scales. At the second-order scale (23-km2 blocks) semi-permanent wetlands were preferred during two winters in which habitat selection could be assessed (1992–1993 and 1994–1995). An additional second-order preference for grass-covered uplands was shown during the mild 1994–1995 winter. At the third-order scale (home-range) pheasants preferred the edges of wetlands in 1992–1993 and 1994–1995. The central portions of wetlands were preferred in 1992–1993 and used proportionately in 1994–1995. Seasonal wetlands were avoided at the third order scale during 1992–1993 and 1994–1995. The average winter survival rate was 0.41, with rates ranging from 0.04–0.86 and differing significantly among winters. Survival was lower during early winter and midwinter periods for birds weighing less than 1090 g and for birds captured in semi-permanent wetlands under private ownership. A 1 C increase in the mean weekly maximum temperature decreased the probability of death by 0.06 and a 2.5 cm increase in new snow raised the probability of death by 0.08

QUANTIFICATION OF CATTAIL (\u3ci\u3eTYPHA\u3c/i\u3e SPP.) IN THE PRAIRIE POTHOLE REGION OF NORTH DAKOTA IN RELATION TO BLACKBIRD DAMAGE TO SUNFLOWER

Sunflower is an important crop for many farmers in the upper Midwest, especially in North Dakota and South Dakota. Blackbirds have been a major problem for the sunflower grower community. Bud depredation to a field can be devastating. The USDA-APHISWS is charged with reducing the conflict between the birds and the farmers. Many methods have been employed by Wildlife Services and other agencies to lessen the damage. One method is the reduction of the cattail (Typha spp.) habitat used by blackbirds in and around wetlands; however, cattails are used by other animals. Consequently, there is a need to insure habitat manipulation is not significantly affecting non-target species, hence knowing what portion of the total cattail habitat is being manipulated is critical. The purpose of this study was to quantify cattail habitat in the Prairie Pothole Region (PPR) of North Dakota. Remote sensing using aerial infrared photographs was used to sample 120, 10.36 km sq. plots, randomly distributed throughout each of four strata dividing the PPR in ND. ArcInfo 8x Geographic Information System (GIs) software was used to run a supervised classification to delineate cattail from other vegetation. Results found 2,245 =t 257 (S.E.) km sq. of cattail in the PPR. These findings show that less than one percent of the total cattail stand in the PPR is being affected by the USDA cattail management efforts

AVIAN USE OF ROADSIDE HABITAT IN THE SOUTHERN DRIFT PLAINS OF NORTH DAKOTA AND IMPLICATIONS FOR CATTAIL (\u3ci\u3eTYPHA\u3c/i\u3e SPP.) MANAGEMENT

We determined avian use of roadside rights-of-way to develop proper management strategies for the manipulation of roadside cattail. Cattail management is a technique used to reduce nesting and roosting habitat for problematic blackbird species, which might feed on sunflower crops in the vicinity of cattail-dominated wetlands. Thirty quarter sections (1 quarter section ≈ 64.75 ha) located in the Southern Drift Plains of North Dakota served as our study units. Roadside habitat along two 0.5 mile (-0.8 km) transects bordering these quarter sections was surveyed to assess avian use. Additionally, nest surveys were , conducted to provide an index of breeding bird use of this roadside habitat. Finally, roadside habitat was surveyed for a number of different habitat variables. We found 49 different species during the surveys. Of the 2,529 birds found in this habitat, 1,479 (41.5%) were blackbirds. Blackbirds were also the primary nesting birds, contributing to 89% of the active nests found in roadside habitat. In terms of avian use and nesting, blackbirds, especially redwings, were the dominant bird species using roadside cattail. With proper management of roadside habitat, potential effects on non-blackbirds can be minimized, and nesting habitat for blackbirds can be reduced. Our data Indicate that a loss in nesting habitat will likely result in fewer blackbirds and a subsequent reduction in sunflower damage