Narrow Results

The work presented in this paper concerns the design of an on-chip GaN transformer. The integrated transformer is composed of two planar stacked coils with spiral octagonal geometry. A comparison is done on different analytical methods to calculate the inductance of the transformer spiral planar coils. Three transformers with different outer diameters are compared to illustrate the influence of the coil geometry. The estimated inductance and dc series resistance are evaluated. Using COMSOL Multiphysics 5.3 software, the thermal effect is illustrated in the integrated transformer operating at high frequencies. The different parasitic effects created by the planar stacked layers are validated by the equivalent electrical circuit, and the different electrical parameters are calculated.

Ordered trees are generally drawn using order-preserving planar straight-line grid drawings. We investigate the area-requirements of such drawings and present several results. Let T be an ordered tree with n nodes. We show that:

• T admits an order-preserving planar straight-line grid drawing with O(n log n) area.

• If T is a binary tree, then T admits an order-preserving planar straight-line grid drawing with O(n loglog n) area.

• If T is a binary tree, then T admits an order-preserving upward planar straight-line grid drawing with optimalO(n log n) area.

We also study the problem of drawing binary trees with user-specified aspect ratios. We show that an ordered binary tree T with n nodes admits an order-preserving planar straight-line grid drawing with area O(n log n), and any user-specified aspect ratio in the range [1,n/log n]. All the drawings mentioned above can be constructed in O(n) time.

Given a set V of n points in a two-dimensional plane, we give an O(nlogn)-time centralized algorithm that constructs a planar t-spanner for V, for such that the degree of each node is bounded from above by , where 0<α<π/2 is an adjustable parameter. Here C_del is the spanning ratio of the Delaunay triangulation, which is at most . We also show, by applying the greedy method in Ref. [14], how to construct a low weighted bounded degree planar spanner with spanning ratio ρ(α)²(1+∊) and the same degree bound, where ∊ is any positive real constant. Here, a structure is called low weighted if its total edge length is proportional to the total edge length of the Euclidean minimum spanning tree of V. Moreover, we show that our method can be extended to construct a planar bounded degree spanner for unit disk graphs with the adjustable parameter α satisfying 0<α<π/3. Previously, only centralized method⁶ of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t≃10.02. The distributed implementation of this centralized method takes O(n²) communications in the worst case. Our method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n).

The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications [7, 8, 11, 12, 13, 17, 18]. The first such algorithms developed were able to traverse triangulated subdivisions [10]. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope [3]. The research progress culminated to an algorithm that can traverse any planar subdivision [6, 9]. In this paper, we extend the notion of planar subdivision to quasi-planar subdivision in which we allow many edges to cross each other. We generalize the algorithm from [9] to traverse any quasi-planar subdivision that satisfies a simple geometric requirement. If we use techniques from [6] the worst case running time of our algorithm is O(|E|log|E|); matching the running time of the traversal algorithm for planar subdivisions [6].

Let G be a finite group and H be a subgroup of G. We introduce the non-normal graph of H in G, denoted by _H,G, and give some of the graph theoretical properties of _H,G.

Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are planar. In addition, we classify the finite groups whose permutability graphs of subgroups are one of outerplanar, path, cycle, unicyclic, claw-free or C₄-free. Also, we investigate the planarity of permutability graphs of subgroups of infinite groups.

Given a group $G$, the enhanced power graph of $G$, denoted by ${{𝒢}}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x$ and $y$ are edge connected in ${{𝒢}}_e(G)$ if there exists $z\in G$ such that $x=z^m$ and $y=z^n$ for some $m,n\in \mathbb{N}$. Here, we show that the graph ${{𝒢}}_e(G)$ is complete if and only if $G$ is cyclic; and ${{𝒢}}_e(G)$ is Eulerian if and only if $\left|G\right|$ is odd. We characterize all abelian groups and all non-abelian $p$-groups $G$ such that ${{𝒢}}_e(G)$ is dominatable. Besides, we show that there is a one-to-one correspondence between the maximal cliques in ${{𝒢}}_e(G)$ and the maximal cyclic subgroups of $G$.

Let $𝕍$ be a k-dimensional vector space over a finite field $𝔽$ with a basis $\{{\alpha }_1,\dots ,{\alpha }_k\}$. The nonzero component graph of $𝕍$, denoted by $\mathrm{\Gamma }(𝕍)$, is a simple undirected graph with vertex set as nonzero vectors of $𝕍$ such that there is an edge between two distinct vertices $x,y$ if and only if there exists at least one ${\alpha }_i$ along which both $x$ and $y$ have nonzero scalars. In this paper, we find the vertex connectivity and girth of $\mathrm{\Gamma }(𝕍)$. We also characterize all vector spaces $𝕍$ for which $\mathrm{\Gamma }(𝕍)$ has genus either 0 or 1 or 2.

2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any .

Let $R$ be a commutative ring with identity. We consider a simple graph associated with $R$, denoted by ${\mathrm{\Omega }}_R^{\ast }$, whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $J\text{Ann}(I)=(0)$ or $I\text{Ann}(J)=(0)$. In this paper, we characterize the commutative Artinian non-local ring $R$ for which ${\mathrm{\Omega }}_R^{\ast }$ has genus one and crosscap one.

Let $R$ be a commutative ring with identity. The cozero-divisor graph of $R$, denoted by ${\mathrm{\Gamma }}^{{}^{\prime }}(R)$, is a graph whose vertex set is $W^{\ast }(R)$, the set of all non-zero and non-unit elements of $R$. Two distinct vertices $a$ and $b$ in $W^{\ast }(R)$ are adjacent if and only if $a\notin Rb$ and $b\notin Ra$. The Crosscap of a graph $G$, denoted by $\overline{\gamma }(G)$, is the minimum integer $k$ such that the graph can be embedded in the non-orientable surface ${{𝒩}}_k$. The planar graph is called $k$-outerplanar if removing all the vertices incident on the outer face yields a $(k-1)$-outerplanar. The Outerplanarity index of a graph $G$ is the smallest $k$ such that $G$ is $k$-outerplanar. In this paper, we characterize the class of rings $R$ (up to isomorphism) for which $0<\overline{\gamma }({\mathrm{\Gamma }}^{{}^{\prime }}(R))\le 2$. Further we characterize all finite rings $R$ (up to isomorphism) for which ${\mathrm{\Gamma }}^{{}^{\prime }}(R)$ has an outerplanarity index two.

Let $R$ be a finite commutative ring with identity, $I$ be an ideal of $R$ and $J(R)$ denotes the Jacobson radical of $R$. The ideal-based zero-divisor graph ${\mathrm{\Gamma }}_I(R)$ of $R$ is a graph with vertex set $V({\mathrm{\Gamma }}_I(R))=\{x\in R-I:xy\in I,$ for some $y\in R-I\}$ in which distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. In this paper, we determine the diameter, girth of ${\mathrm{\Gamma }}_{J(R)}(R)$. Specifically, we classify all finite commutative nonlocal rings for which ${\mathrm{\Gamma }}_{J(R)}(R)$ is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of ${\mathrm{\Gamma }}_{J(R)}(R)$ and characterize all of them.

Planar electrode array is an important tool to evaluate perceptual or cognitive functions of the cortex and prosthetic applications. Many construction methods have been developed. To maximize the usefulness of an array electrode, a low-cost, precise, and flexible microelectrode array with low man power and short construction duration is crucial. In this study, we introduced an 8 × 8 microelectrode array on a flexible polyimide film through microelectronics fabrication. The array dimension was capable of covering the primary somatosensory cortex of a rat. The microelectrode array was insulated with biocompatible Parylene-C except of microelectrode tip. Each electrode tip was 66 μm height and separated with 0.5 mm to refine a detail somatic sensory processing. In pentobarbital anesthetized rats, stable spontaneous brain activity was successfully recorded through the electrode array. In addition, positive peaks of somatosensory evoked potentials (SEPs) elicited by stimulating rat's whisker pad, forepaw, hindpaw, and tail were obviously and consistently recorded. Latencies of SEPs increased as caudal part of the body was stimulated. The SEPs from stimulation of 4 body parts revealed different spatiotemporal patterns, which indicated a somatotopic organization of the rat. Our results demonstrated the superiority of the planar microelectrode array on the application of simultaneous recording and analysis of the brain activity in rats.