The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Phase Role in the Non-Uniformity of Main-Line Couplings in Asymmetric Extracted-Pole Inline Filters

The use of classical symmetrical polynomial definition to synthesize fully canonical inline filters with an asymmetrical distribution of the transmission zeros along the topology leads to the occurrence of uneven admittance inverter in the main-line. This form introduces some limitations to transform such topology into a ladder network. Despite circuital transformation can be used to accommodate both technology and topology, it is usual that extra reactive elements are necessary to implement phase shifts required to achieve the complete synthesis. This article introduces a novel method able to determine the required phase correction that has to be applied to the characteristic polynomials in order to equalize all the admittance inverters in the main path to the same value. It has been demonstrated that a suitable pair of phase values can be accurately estimated using a developed hyperbolic model which can be obtained from the transmission and reflection scattering parameters. To experimentally validate the proposed method, a Ladder-type filter with asymmetrical polynomial definition has been synthesized, fabricated, and measured, demonstrating the effectiveness of the developed solution.

Stability Test of Networked Systems with Multiple Delays

In this paper, we propose a sufficient stability condition for networked systems with multiple delays based on the 2-D polynomials and 2-D Hurwitz-Schur stability. The main advantage of the new stability condition is that it is applicable to the general case of networked systems with multiple, incommensurate delays yet numerically tractable. The characteristic polynomials of networked systems are mapping into 2-D hybrid polynomials, then to test the Hurwitz-Schur stability can. determine the networked systems, examples including system simulations verify the validity of the proposed test algorithms.

Formation of Characteristic Polynomials on the Basis of Fractional Powers j of Dynamic Systems and Stability Problems of Such Systems

The article presents the creation of characteristic polynomials on the basis of fractional powers j of dynamic systems and problems related to the determination of the stability intervals of such systems.

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.