Pair Sum Research Articles

BVP: Balanced Vehicular Pairing for Fair Resource Distribution in Downlink NOMA

In this paper, a vehicular pairing scheme is proposed to balance the resource distribution in non-orthogonal multiple access (NOMA) based vehicular networks. The proposed BVP: balanced vehicular pairing scheme chooses pairs based on fair inter-vehicle distances through the base station (BS) for mobile NOMA users with high mobility. BVP reduces the biased pairing and the disparity of resource utilization by each pair, resulting in fewer signal decoding errors than the existing state-of-the-art solutions. The exhaustive evaluation in Matlab-based network simulator, using real vehicular traffic traces from OpenStreetMap (OSM) and simulation of urban mobility (SUMO) traffic simulator shows that the proposed BVP outperforms existing schemes. The proposed solution reduces throughput and pair sum rate standard deviation by 2 ~ 3 b/s/Hz for every device and every resource, while also significantly reducing the decoding bit error rate.

Edge pair sum labeling of some classes of graphs

In 2013, we [3] introduced the concept of edge pair sum labeling and further studied in [5-15]. In this paper, we find some new results on edge pair sum labeling.

A New Signal Representation Using Complex Conjugate Pair Sums

This letter introduces a real valued summation known as Complex Conjugate Pair Sum (CCPS). The space spanned by CCPS and its one circular downshift is called {\em Complex Conjugate Subspace (CCS)}. For a given positive integer $N\geq3$, there exists $\frac{\varphi(N)}{2}$ CCPSs forming $\frac{\varphi(N)}{2}$ CCSs, where $\varphi(N)$ is the Euler's totient function. We prove that these CCSs are mutually orthogonal and their direct sum form a $\varphi(N)$ dimensional subspace $s_N$ of $\mathbb{C}^N$. We propose that any signal of finite length $N$ is represented as a linear combination of elements from a special basis of $s_d$, for each divisor $d$ of $N$. This defines a new transform named as Complex Conjugate Periodic Transform (CCPT). Later, we compared CCPT with DFT (Discrete Fourier Transform) and RPT (Ramanujan Periodic Transform). It is shown that, using CCPT we can estimate the period, hidden periods and frequency information of a signal. Whereas, RPT does not provide the frequency information. For a complex valued input signal, CCPT offers computational benefit over DFT. A CCPT dictionary based method is proposed to extract non-divisor period information.

Some results on super pair sum graphs

Let G be a graph with P vertices and q edges. The graph G is said to be a super pair sum labeling if there exists a bijection f from when p + q is odd and from when p + q is even such that f(uv) = f(u) + f(v). A graph that admits a super pair sum labeling is called a super pair sum graph. In this paper, we prove that the graphs Hn ⊙ mK1, (P2n; Sm), S’(P2n), < Bm,n : Pk > for m ≥ 1, n ≥ 1, k ≡ 2(mod 4), < B(m): Pk > for m ≥ 1, k ≡ 0, 2(mod 4) and 2Bm,n (m ≥ 1, n ≥ 1) are super pair sum graphs.

Edge pair sum labeling of some cartesian product of graphs

Let [Formula: see text] be a [Formula: see text] graph. An injective map [Formula: see text] is said to be an edge pair sum labeling if the induced vertex function [Formula: see text] defined by [Formula: see text] is one-to-one where [Formula: see text] denotes the set of edges in [Formula: see text] that are incident with the vertex [Formula: see text] and [Formula: see text] is either of the form [Formula: see text] or [Formula: see text] accordingly as [Formula: see text] is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper, we prove that the graphs [Formula: see text], [Formula: see text], book graph and [Formula: see text] admit edge pair sum labeling.

Edge pair sum labeling of butterfly graph with shell order

The concept of an edge pair sum labeling was introduced in [3]. Let $G(p, q)$ be a graph. An injective map $f: E(G) \rightarrow\{ \pm 1, \pm 2, \cdots, \pm q\}$ is said to be an edge pair sum labeling if the induced vertex function $f^*: V(G) \rightarrow Z-\{0\}$ defined by $f^*(v)=\sum_{e \epsilon E_v} f(e)$ is one- one where $E_v$ denotes the set of edges in $G$ that are incident with a vertex $v$ and $f^*(V(G))$ is either of the form $\left\{ \pm k_1, \pm k_2, \cdots, \pm k_{\frac{p}{2}}\right\}$ or $\left\{ \pm k_1, \pm k_2, \cdots, \pm k_{\frac{p-1}{2}}\right\}$ $\cup\left\{ \pm k_{\frac{p+1}{2}}\right\}$ according as $p$ is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the shell graph and butterfly graph with shell order are edge pair sum graphs.

Lattice 3-Polytopes with Six Lattice Points

We classify lattice $3$-polytopes of width larger than one and with exactly $6$ lattice points. We show that there are $74$ polytopes of width $2$, two polytopes of width $3$, and none of larger width. We give explicit coordinates for representatives of each class, together with other invariants such as their oriented matroid (or order type) and volume vector. For example, according to the number of interior points these $76$ polytopes divide into $23$ tetrahedra with two interior points (clean tetrahedra), $49$ polytopes with one interior point (the $49$ canonical three-polytopes with five boundary points previously classified by Kasprzyk) and only $4$ hollow polytopes. We also give a complete classification of three-polytopes of width one with $6$ lattice points. In terms of the oriented matroid of these six points, they lie in eight infinite classes and twelve individual polytopes. Our motivation comes partly from the concept of distinct pair sum (or dps) polytopes, which, in dimension $3$, can have at most $8$ lattice points. Among the $74+2$ classes mentioned above, exactly $44 + 1$ are dps.

Some Edge Pair Sum Graphs

An injective map f : E(G)→{± 1,±2, … ,±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f* : V(G)→Z − {0} defined by f* (v)=∑e∈Evf(e) is one – one, where Ev denotes the set of edges in G that are incident with a vertex v and f*(V(G)) is either of the form or according as p is even or odd. A graph which admits edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the one point union of cycles, the perfect binary tree, shadow graph, total graph and Pn2 admit edge pair sum labeling.

Some results on edge pair sum labeling

An injective map f:E(G)→{±1,±2,⋯,±q} is said to be an edge pair sum labeling if the induced vertex function f⁎:V(G)→Z−{0} defined by f⁎(v)=∑eϵEvf(e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f⁎(V(G)) is either of the form {±k1,±k2,⋯,±kp2} or {±k1,±k2,⋯,±kp−12}⋃{kp+12} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn⋃Cn and K1,n⋃K1,m are edge pair sum graphs.

Edge Pair Sum Labeling

An injective map f : E(G) ? {±1, ±2, ,±q } is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f*:V(G) ? Z – {0} defined by f*(?) = ?e?E? f(e) is one-one, where denotes the set of edges in G that are incident with a vertex v and f*(V(G)) is either of the form {±k1, ±k2, , ±kp/2} or {±k1, ±k2, , ±k(p-1)/2}U{kp/2}according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that path Pn, cycle Cn, triangular snake, PmUK1,n, Cn?Kmc are edge pair sum graphs. Keywords: Pair sum graph, edge pair sum labeling, edge pair sum graph.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i3.15001 J. Sci. Res. 5 (3), 457-467 (2013)

Pair Sum Labeling of some Special Graphs

Pair Sum Labeling of some Special Graphs

Further Results on Pair Sum Graphs

Let G be a graph. An injective map is called a pair sum labeling if the induced edge function, defined by is one-one and is either of the form or according as q is even or odd. A graph with a pair sum labeling is called a pair sum graph. In this paper we investigate the pair sum labeling behavior of subdivision of some standard graphs.

A Note on Pair Sum Graphs

Let G be a (p,q) graph. An injective map ƒ: V (G) →{±1, ±2,...,±p} is called a pair sum labeling if the induced edge function, ƒe: E(G)→Z -{0} defined by ƒe (uv)=ƒ(u)+ƒ(v) is one-one and ƒe(E(G)) is either of the form {±k1, ±k2,…, ±kq/2} or {±k1, ±k2,…, ±k(q-1)/2} {k (q+1)/2} according as q is even or odd. Here we prove that every graph is a subgraph of a connected pair sum graph. Also we investigate the pair sum labeling of some graphs which are obtained from cycles. Finally we enumerate all pair sum graphs of order ≤ 5.Keywords: Cycle; Path; Bistar; Complete graph; Complete bipartite graph; Triangular snake.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i2.6290 J. Sci. Res. 3 (2), 321-329 (2011)

Further Results on Pair Sum Labeling of Trees

Let G be a (p,q) graph. An injective map f:V(G)→﹛±1,±2,...±p﹜is called a pair sum labeling if the induced edge function, fe:E(G)→Z-﹛0﹜defined by fe(uv)=f(u)+f(v) is one-one and fe(E(G)) is either of the form {±k1,±k2,...,±kq/2} or {±k1,±k2,...,±kq-1/2}∪{kq+1/2} according as q is even or odd. A graph with a pair sum labeling is called a pair sum graph. In this paper we investigate the pair sum labeling behavior of some trees which are derived from stars and bistars. Finally, we show that all trees of order nine are pair sum graphs.

Magic square spectra

To study the eigenvalues of low order singular and non-singular magic squares we begin with some aspects of general square matrices. Additional properties follow for general semimagic squares (same row and column sums), with further properties for general magic squares (semimagic with same diagonal sums). Parameterizations of general magic squares for low orders are examined, including factorization of the linesum eigenvalue from the characteristic polynomial. For nth order natural magic squares with matrix elements 1 , … , n 2 we find examples of some remarkably singular cases. All cases of the regular (or associative, or symmetric) type (antipodal pair sum of 1 + n 2 ) with n - 1 zero eigenvalues have been found in the only complete sets of these squares (in fourth and fifth order). Both the Jordan form and singular value decomposition (SVD) have been useful in this study which examines examples up to 8th order. In fourth order these give examples illustrating a theorem by Mattingly that even order regular magic squares have a zero eigenvalue with odd algebraic multiplicity, m. We find 8 cases with m = 3 which have a non-diagonal Jordan form. The regular group of 48 squares is completed by 40 squares with m = 1 , which are diagonable. A surprise finding is that the eigenvalues of 16 fourth order pandiagonal magic squares alternate between m = 1 , diagonable, and m = 3 , non-diagonable, on rotation by π / 2 . Two 8th order natural magic squares, one regular and the other pandiagonal, are also examined, found to have m = 5 , and to be diagonable. Mattingly also proved that odd order regular magic squares have a zero eigenvalue with even multiplicity, m = 0 , 2 , 4 , . . . Analyzing results for natural fifth order magic squares from exact backtracking calculations we find 652 with m = 2 , and four with m = 4 . There are also 20, 604 singular seventh order natural ultramagic (simultaneously regular and pandiagonal) squares with m = 2 , demonstrating that the co-existence of regularity and pandiagonality permits singularity. The singular odd order examples studied are all non-diagonable.

A pair sum approximation for predicting defect cluster binding energies in fluorites

Atomic scale computer simulations have been used to investigate clustering of those defects that result from the accommodation of Gd2O3 in a CeO2 host lattice. It has been found that binding energies of the defect clusters can be accurately described by a model in which the total binding energy is deconvoluted into its constituent pair terms (though a three body correction is necessary for a specific cluster type). At short defect–defect distances, these interaction energies must be calculated explicitly because of the complexity of the lattice relaxation. At longer distances, binding energies can be approximated by calculating the coulomb interactions between the effective charges of the defects divided by the relative permittivity of the host lattice. This scheme provides a framework within which the total of defect–defect interactions can be determined for a large ensemble of defects disordered throughout this lattice.

Discovery of a Galaxy Cluster in the Foreground of the Wide‐Separation Quasar Pair UM 425

We report the discovery of a cluster of galaxies in the field of UM 425, a pair of quasars separated by 65. Based on this finding, we revisit the long-standing question of whether this quasar pair is a binary quasar or a wide-separation lens. Previous work has shown that both quasars are at z = 1.465 and show broad absorption lines. No evidence for a lensing galaxy has been found between the quasars, but there are two hints of a foreground cluster: diffuse X-ray emission observed with Chandra, and an excess of faint galaxies observed with the Hubble Space Telescope. Here we show, via VLT spectroscopy, that there is a spike in the redshift histogram of galaxies at z = 0.77. We estimate the chance of finding a random velocity structure of such significance to be about 5%, and thereby interpret the diffuse X-ray emission as originating from z = 0.77 rather than from the quasar redshift. The mass of the cluster, as estimated from either the velocity dispersion of the z = 0.77 galaxies or the X-ray luminosity of the diffuse emission, would be consistent with the theoretical mass required for gravitational lensing. The positional offset between the X-ray centroid and the expected location of the mass centroid is ~40 kpc, which is not too different from offsets observed in lower redshift clusters. However, UM 425 would be an unusual gravitational lens, by virtue of the absence of a bright primary lensing galaxy. Unless the mass-to-light ratio of the galaxy is at least 80 times larger than usual, the lensing hypothesis requires that the galaxy group or cluster plays a uniquely important role in producing the observed deflections.