A Note on the Matching Polynomials of Paths and Cycles

The spectra of matching polynomials which are useful in the computations of resonance energy and grand canonical partition functions of molecular's. It also present other properties for certain classes of graphs and lattices. In [1] Balasubramanian calculates several matching polynomials and matching roots of several molecular graphs. He found that the matching polynomial of C_6, C_(10), C_(14), C_(18) and C_(22) are divided by x^2-2. In this note,we prove that x^2-2 divides MC_(4k+2)(x), k = 1, 2,..., n and obtain some other properties of matching polynomials of paths and cycles.

The Matching Equivalence Graphs with the Maximum Matching Root Less than or Equal to 2

In the paper, we give a necessary and sufficient condition of matching equivalence of two graphs with the maximum matching root less than or equal to 2.

The Number of Matching Equivalent for the Union Graph of Vertices and Cycles

For two graphs G and H, if G and H have the same matching polynomial, then G and H are said to be matching equivalent. We denote by δ (G), the number of the matching equivalent graphs of G. In this paper, we give δ (sK1 ∪ t1C9 ∪ t2C15), which is a generation of the results of in [1].

The Minimum Hosoya Index of a Kind of Tetracyclic Graph

Let be a graph with n vertices and m edges. The sum of absolute value of all coefficients of matching polynomial is called Hosoya index. In this paper, we determine 2nd to 4th minimum Hosoya index of a kind of tetracyclic graph, with m = n +3.

Uniform supertrees with extremal spectral radii

A supertree is a connected and acyclic hypergraph.We investigate the supertrees with the extremal spectral radii among several kinds of r-uniform supertrees.First,by using the matching polynomials of supertrees,a new and useful grafting operation is proposed for comparing the spectral radii of supertrees,and its applications are shown to obtain the supertrees with the extremal spectral radi among some kinds of r-uniform supertrees.Second,the supertree with the third smallest spectral radius among the r-uniform supertrees is deduced.Third,among the r-uniform supertrees with a given maximum degree,the supertree with the smallest spectral radius is derived.At last,among the r-uniform starlike supert rees,the supertrees with the smallest and the largest spectral radii are characterized.

A COMPLETE SOLUTION OF HOSOYA'S MYSTERY

In this note a theorem concerning the coincidence between the characteristic polynomial of a cycle and the polynomial of Kekule structure count of a primitive coronoid is presented which implies a complete solution of Hosoya's mystery.

Neurocomputing van der Pauw function for the measurement of a semiconductor's resistivity without use of the learning rate of weight vector regulation

Van der Pauw＇s function is often used in the measurement of a semiconductor＇s resistivity. However, it is difficult to obtain its value from voltage measurements because it has an implicit form. If it can be expressed as a polynomial, a semiconductor＇s resistivity can be obtained from such measurements. Normally, five orders of the abscissa can provide sufficient precision during the expression of any non-linear function. Therefore, the key is to determine the coefficients of the polynomial. By taking five coefficients as weights to construct a neuronetwork, neurocomputing has been used to solve this problem. Finally, the polynomial expression for van der Pauw＇s function is obtained.