Here, we analyze and compare properties of Cayley graphs of permutation graphs called transposition graphs as this family of graphs has better degree and diameter properties than the hypercube. First, the conjugacy class of the permutation group combined with its generators dictates the type of symmetry possessed by their respective Cayley graphs. In addition, the base-b hypercube of dimension n is a class of networks known to possess virtually every known notion of symmetry. Second, properties of the transposition graphs are given. Third, the spectrum of this class of graphs is discussed and formulas for the energy of these graphs are given as they are dependent on these properties. DOI: 10.1007/978-3-031-05375-7_23

Among the challenges terrapins encounter are habitat loss due to coastal development and sea-level rise, mortality at all life stages by mammalian and avian predators, road mortality, boat strikes, harvest for the pet trade, and drowning in crab traps. The primary objective of this study was to locate populations and nesting areas of diamondback terrapins in the four northeastern-most counties of Florida (Nassau, Duval, St. Johns, and Flagler). We collected 404 records of terrapin activity in 2013 and 2014. Most were from Nassau County (277) and only one was from Flagler County. Most data were in the form of depredated nests (205) and terrapin remains (147). The woody plant data suggest that terrapins were significantly more likely to nest when Christmas berry (Lycium carolinianum) was present, and nesting was less likely when either wax myrtle (Myrica cerifera) or oak (Quercus spp.) were present.

There were 83,083 knee arthroscopic procedures between 2012 and 2017 obtained from the National Surgical Quality Improvement Program database. The overall readmission rate was 0.87%. The complication rates were highest for synovectomy and cartilage procedures, 1.6% and 1.3% respectively. A majority of readmissions were related to the procedure (71.1%) with wound complications being the primary reason (28.2%) followed by pulmonary embolism and deep vein thrombosis, 12.7% and 10.6%, respectively. Gender and body mass index were not significant factors and age over 65 years was an independent risk factor. Wound infection, deep vein thrombosis, and pulmonary embolism were the most prevalent complications.

Using spectral frequency analysis and visualization techniques on pediatric electroencephalogram recordings, we identify a method of detecting seizures thirty seconds before seizure onset by identifying significant preictal locations and their respective frequencies within the gamma band of 30 through 100 Hz. Using average log-power differences and visualization techniques to identify significant preictal patterns, intractable seizures are found to have common frequency extremes (CFEs) in the high gamma band between 70 and 100 Hz. Using this data, machine learning detection algorithm predictive performance may be improved by incorporating high gamma band signal processing at the varying location(s) and strength of a pediatric patient’s CFEs.
DOI: 10.1109/ACCESS.2021.3087782

Classically, the heat equation on bounded domains, specifically on tori, yield theta functions as solutions. We study a finite analogue of the Poincare upper half-plane, namely, the finite upper half-plane introduced by Terras. Using this case, we investigate the periodicity inherent in the solutions of bounded domains. The solutions involve zonal spherical functions which come with a natural periodicity. In addition, the related theta functions are automorphic forms. The resultant periodicity interweaves representation theory with the heat equation. We hope this paper stimulates more study of this interplay.
DOI: 10.1090/proc/15610

In this paper, we find another bound on the energy of all graphs including bipartite graphs using spectral moments and properties of cubics. The goal is to find bounds on the energy E(G) of graphs by using the second and fourth spectral moments, M₂ and M₄ using Lagrange multipliers.

To further understand this extremely poor prognosis disease, we evaluated the effect of the treatment facility volumes on overall survival (OS) over the years, especially after the approval of multimodality therapy using temozolomide (TMZ) in 2005. National Cancer Data Base (NCDB) was utilized to identify GBM cases from 2004 through 2013 using ICD-O-3 code 9440/3 to identify eligible patients. We focused on studying the association between treatment facility volume and OS after adjusting for the patient-, disease-, and facility-characteristics.

A total of 60,672 eligible GBM patients with median age of 65 years, treated at 1166 facilities were included in this analysis. The median annual facility volume was 3 patients/year (range: 0.1–55.1) and median OS was 8.1 months. There was an improvement in OS across all facilities after 2005, when multimodality therapy with TMZ was approved. Treatment at quartile 4 centers (Q4; >7 patients/year) was independently associated with decreased all-cause mortality in a multivariate analysis (Q3 hazard ratio [HR]: 1.11, 95% CI 1.09, 1.13; Q2 HR: 1.15, 95% CI 1.12, 1.19; Q1 HR: 1.25, 95% CI 1.17, 1.33). Treatment facility volume independently affects OS among GBM patients. Factors that are variable in high- and low-volume centers should be addressed to mitigate outcome disparities.

with Sonikpreet Aulakh, MD, Joseph Free, Aneel Paulus MD, Steven Rosenfeld MD,PhD, Alfredo Quinones-Hinojosa MD, Alak Manna PhD, Rami Manochakian MD, Asher Chanan-Khan MD, J Clin Neurosci. 2019 Oct; 68:271-274. doi: 10.1016/j.jocn.2019.04.028

This paper determines if certain families of Heisenberg graphs satisfy the Ramanujan bound. Proving conditions for Ramanujancy and non-Ramanujancy is desirable as a Ramanujan graph can represent efficient communication networks in that they minimize cost of wiring and maintenance, but maximize the number of connections from one vertex to another. We say that a graph is Ramanujan if it is k-regular, simple, connected, undirected and the largest nontrivial eigenvalue, λ, of its adjacency matrix satisfies the condition that |λ|≤sqrt(2k-1) where k is the degree of the graph. We find λ by using the exponential sum lemma and prove a general theorem for the non-Ramanujancy of these graphs under certain conditions. One of these conditions is satisfied by assuming the minimality of certain terms in the exponential sum. Then, without loss of generality, we isolate and prove the conditions for the occurrence of all lower-dimensional eigenvalues that exceed the Ramanujan bound. Also, graphical analysis of the spectra of Ramanujan graphs is performed and it is shown that, for a fixed number of elements in the symmetric set and as a prime p increases without bound, the distribution of the eigenvalues resembles the Sato-Tate semi-circle distribution.

with Vincent Dang and Yang Ge, Proceedings of the Thirty-Eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 185 (2007), 111–126. MR2408803 05C50

On the Ramanujancy of Heisenberg Graphs of degree 6 or more

For q = p, the graph G^∗(q) is called the q^th Platonic graph. The name comes from the fact that when p = 3 or 5, the graphs G^∗(p) correspond to 1-skeletons of two of the Platonic solids, in other words, the tetrahedron and the icosahedron as in [2]. More generally, G^∗(p) is the 1-skeleton of a triangulation of the modular curve X(p). In this paper, we determine the spectrum of G^∗(q) and thus generalize results of P.E. Gunnells, “Some elementary Ramanujan graphs”.

with Dominic Lanphier and Marvin Minei, Discrete Math. 307 (2007), no. 9-10, 1074–1081.  DOI: 10.1016/j.disc.2006.07.032   MR2292536 05C25 (05C50) 

This paper continues the authors previous work that deals with the problem of reconstructing some function defined on a finite group G from averages of the function over various neighborhoods of points. It generalizes work done earlier by Diaconis and Graham, and Fill, among others.

Diaconis and Graham discuss the cases where G = Z^k₂, the group of binary k-tuples, and where G = S_n, the symmetric group on n letters, in order to provide an exposition on discrete Radon transforms which appear in applied statistics. Fill examines the case G = Z^k₂, the group of integers modulo n. This case arises in directional data analysis and circular time series. Other finite analogues of the Radon transform occur, but we shall restrict ourselves to the case G = Z^k_n, the group of k-tuples of the integers modulo n, as these results are of use in k-dimensional toroidal time series.

with Elinor Velasquez. SIAM J. Discrete Math. 18 (2004/05), no. 3, 472–478. MR2134409 44A12  (43A75)
DOI: 10.1137/S0895480103430764

The aim of this paper is to study the special functions known as Ihara Selberg zeta functions for Cayley graphs of finite Heisenberg groups as well as their factorizations into products of Artin-Ihara L-functions. The Heisenberg group H(R) over a ring R consists of upper triangular 3×3 matrices with entries in R and ones on the diagonal. The Ihara-Selberg zeta function is analogous to the Riemann zeta function with primes replaced by certain closed paths in a graph. This paper is a continuation of (DeDeo et al., 2004) where we presented a study of the statistics of the spectra of adjacency matrices of finite Heisenberg graphs.

with Martínez M., Medrano A., Minei M., Stark H., Terras A. (2005) Zeta Functions of Heisenberg Graphs over Finite Rings. In: Ismail

DOI: 10.1007/0-387-24233-3_8     MR2132463     05C25 (05C38 11M41 11R42 20H25) 

In this paper, we study the dual transformation applied to N particles and N −1 springs that are no longer constrained to the real line. For simplicity, assume the Hamiltonian system formed to be embedded in Euclidean n−space with the standard metric and also assume that N < ∞ and that all N particles are connected to at least one spring.

Our interpretation of one possible extension is to create graphs that can act as dynamical systems. We decide to take a graph (a finite one, for simplicity), position masses at the vertices of the graph, and exchange the edges of the graphs with springs. Again, for simplicity, we assume that the springs behave uniformly, although not necessarily linearly. We then re-express the Hamiltonian with respect to the graphs rather than focusing on the underlying manifold. We focus on graphs which may be neatly embedded into Euclidean n−dimensional space for the purpose of this paper. Hence we now have information regarding how graphs behave when “equipped” with a symplectic 2−form.

with Elinor Velasquez. Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium 169 (2004), 127–139.   MR2122063     70H06 (05C45 37J35 37K60 70H15 82C05)

An Introduction to Symplectic Maps and Generalizations of the Toda Lattice

Although Kloosterman sums of odd prime powers have been studied extensively, many authors avoid the case where p = 2. This case is important as Kloosterman sums for fields of characteristic 2 have been useful in error-correcting codes and Kloosterman sums over the ring of integers modulo pr have been helpful in studying the spectra of Euclidean graphs. After stating some well-known facts about Kloosterman sums, we evaluate several generalized Kloosterman sums modulo 2^r.

Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 165 (2003), 65–75. MR2049122 11T24 (11L05)

Generalized Kloosterman sums over Rings of Order 2^r

The aim of this paper is to study spectra of Cayley graphs attached to finite Heisenberg groups using histograms and Hofstadter butterfly figures. The Heisenberg group H(R) over a ring R consists of upper triangular 3×3 matrices with entries in R and ones on the diagonal.

Heisenberg groups over finite fields have provided a tool in the search for random number generators (see Maria Zack) as well as the search for Ramanujan graphs (see Perla Myers). Other references are P. Diaconis and L. Saloff-Coste [5] and Terras [24]. As shown in the last reference, the size of the second largest (in absolute value) eigenvalue of the adjacency matrix governs the speed of convergence to uniform for the standard random walk on a connected regular graph. Ramanujan graphs have the best possible eigenvalue bound for connected regular graphs of fixed degree in an infinite sequence of graphs with number of vertices going to infinity. For such graphs, the random walker gets lost as quickly as possible. Equivalently, this says that such graphs can be used to build efficient communication networks.

with M. Martinez, A. Medrano, M. Minei, H.M. Stark, and A. Terras. Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dyn. Syst. 2003, suppl., 213–222. MR2018119 11T60 (05C25)

https://www.aimsciences.org/article/doi/10.3934/proc.2003.2003.213