Energy of a graph is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix A(G) of a graph G and is denoted by E(G). The graph G with n vertices is called nonhypoenergetic if E(G) ≥ n and hypoenergetic if E(G) < n. Singular graphs are graphs with nullity η(G) > 0. In this paper we study about nonhypoenergetic and hypoenergetic graphs among singular graphs. We also construct singular graphs of larger nullity using coalescence. c
This paper studies the characteristic polynomial of distance matrices and adjacency matrices of some families of weakly semi-regular bipartite graphs. The D n,r decomposition is a decomposition of distance matrices and adjacency matrices of some families of graphs. The general distance matrices and adjacency matrices of these graphs are decomposed into a further simpler form. The spectra of these graphs are also analysed. The D n,r decomposition of these graphs is done so that analysis of eigenvalues and related parameters are possible with this decomposition. The heat kernel of these graphs is analysed.
This paper studies the correlation properties of a family of product codes, obtained as the product of two distinct rn-sequences. First we define a set of primary product codes in terms of the two rn-sequences. The collection of the autocorrelation functions of the primary product codes are divided into ensembles. It is shown that these ensembles are cyclic shifted versions of the crosscorrelation function between the two rn-sequences that generate the product code. The relationships between these ensembles are also derived. The autocorrelation histograms of a set of primary product codes corresponding to an integer class are shown to be equal. For the special case of product codes referred to as the Cold codes, obtained as the product of two preferred rn-sequences, the amount of unbalance in the autocorrelation histograms of the primary Gold codes are related to the cross-correlation between the preferred rn-sequences. IntroductionProduct codes such as the Gold codes have found a variety of applications in spread spectrum multiple access (ssMA) communications. A family of product codes consists of the code sequences that are obtained as the product of two maximal length, linear feedback shift register (LFSR) generated sequences (rn-sequences). The performance during acquisition of a SSNA transmitter signal by a SSMA receiver depends on the histogram of the autOcorrelation function of the product code family member used by the transmitter. Knowledge of the. histograms of the autocorrelation behavior of these codes will facilitate the optimal selection of codes, design of acquisition strategy 1] and performance analysis of SSMA systems. This paper relates the autocorrelation function and its histogram of a product code member to the cross-correlation between the two rn-sequences that generate the product code. The histograms remain the same for a group of product code members that are related by an integer class. The ensemble of autocorrelation values of the primary product code members corresponding to a fixed delay is shown to be a cyclic shifted version of the crosscorrelation between the two characteristic rn-sequences that generate the product code. The relationship between these cyclic shifts for delays belonging to an autocorrelat ion residue class is presented. The above properties enable a set of numbers, orders of magnitude fewer than the actual autocorrelation functions, to summarize the autocorrelation values of all members of a product code for all delays. For the special class of product codes referred to as Gold codes, the autocorrelation histogram can also be related to the crosscorrelation between the preferred rn-Sequences that generate the Gold code family. Several examples are presented to illustrate the properties.Section two reviews the basics in rn-sequences.The third section defines a family of product codes, discusses its autocorrelation properties and autocorrelation histograms. The last section introduces the Gold codes, discusses its autocorrelation behavior and autocorrelation histogra...
A singular graph G has an adjacency matrix A(G) with nullity η(G) > 0. Vertices of singular graphs are classified as core and noncore vertices. There are two types of noncore vertices: noncore vertices of zero null spread and of null spread −1. Deletion of these vertices from a singular graph either changes the nullity or leave it unaltered. In this paper larger singular and nonsingular graphs were constructed by joining singular graphs by a path. As singular graphs have different types of vertices, the graphs constructed in this way differ in nullity depending on the vertex we are joining during construction. An attempt was made to construct singular graph of maximum nullity. Various spectral properties of the resulting graphs were studied.
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