Groups all of whose undirected Cayley graphs are integral

Let $G$ be a finite group, $S\subseteq G\setminus\{1\}$ be a set such that if $a\in S$, then $a^{-1}\in S$, where $1$ denotes the identity element of $G$. The undirected Cayley graph $Cay(G,S)$ of $G$ over the set $S$ is the graph whose vertex set is $G$ and two vertices $a$ and $b$ are adjacent whenever $ab^{-1}\in S$. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group $G$ Cayley integral whenever all undirected Cayley graphs over $G$ are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing $4$ or $6$. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group $S_{3}$ of degree $3$, $C_{3} \rtimes C_{4}$ and $Q_{8}\times C_{2}^{n}$ for some integer $n\geq 0$, where $Q_8$ is the quaternion group of order $8$.Comment: Title is change

Neumaier Cayley graphs

A Neumaier graph is a non-complete edge-regular graph with the property that it has a regular clique. In this paper, we study Neumaier Cayley graphs. We give a necessary and sufficient condition under which a Neumaier Cayley graph is a strongly regular Neumaier Cayley graph. We also characterize Neumaier Cayley graphs with small valency at most $10$.Comment: 17 pages, 1 figur

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most $4$, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency $5$, and the Cayley graphs among all distance-regular graphs with girth $3$ and valency $6$ or $7$. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

Distance-regular Cayley graphs with small valency

We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most 4, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 5, and the Cayley graphs among all distance-regular graphs with girth 3 and valency 6 or 7. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some “exceptional” distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs

Modeling of Short-Channel Effects in GaN HEMTs

In this article, we propose an explicit and analytic charge-based model for estimating short-channel effects (SCEs) in GaN high-electron-mobility transistor (HEMT) devices. The proposed model is derived from the physical charge-based core of the ecole Polytechnique Federale de Lausanne (EPFL) HEMT model, which treats HEMT as a generalized MOSFET. The main emphasis of this article is to estimate SCEs by effectively capturing 2-D channel potential distribution to calculate the reduced barrier height, drain-induced barrier lowering (DIBL), velocity saturation, and channel length modulation (CLM). The model is validated with TCAD simulation results and agreed with measurement data in all regions of operation. This represents the main step toward the design of high-frequency and ultralow-noise HEMT devices using AlGaN/GaN heterostructures

On bipartite distance-regular Cayley graphs with small diameter

We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavi\v{c} and Poto\v{c}nik that relates difference sets to bipartite distance-regular Cayley graphs with diameter three to the case of diameter four. This new case involves certain partial geometric difference sets and -- in the antipodal case -- relative difference sets.Comment: 21 page