Acyclic polynomial

Closely related to the Z-counting polynomial, the matching polynomial (or acyclic polynomial... 

Ivanciuc, O. (1998d) Chemical graph polynomials. Part 4. Non-isomorphic graphs with identical acyclic polynomials. Rev. Roum. Chim., 43, 1173-1179. 

The edge cut sets are of major importance for certain expansions of the characteristic and acyclic polynomials of graphs (see Section 9) as well as for the discussion of some topolo cal properties (see Section 12). 

The matching polynomial Ma(G,x) was introduced by Gutman et al [68-71] in connection with the theory of the so-called topolofpcal resonance energy (see later). In these papers Ma(G,x) was referred to as the "acyclic polynomial . Independently, Aihara [72] develc ed a fully equivalent resonance energy concept he named Ma(( ,x) the "reference p ynomial . 

Encouraged by the success of the Hess—Schaad approach on the Hiickel level, a topological approach was advanced by the Zagreb group " and by Aihara. It is based on the computation of an acyclic polynomial in the framework of graph theory. This is used to describe the acyclic polyene-like reference structure. The topological resonance energy (TRE) is defined as... 

Here TV is the number of vertices in a graph (which corresponds to the number of atoms in a conjugated molecule), Xj are the roots of the characteristic polynomial of the aromatic system, and xf are the roots of the acyclic polynomial of the polyene-like reference system. In essence this corresponds to the procedure of the Hiickel method to solve for the eigenvalues Xj of the Hiickel matrix in units of Finally, gj is the orbital occupancy number. The method was applied to a large number of conjugated hydrocarbons - with results for TREPE that usually show a similar trend as the HSREPE values. Aiha-ra76.77 extended the concept to three-dimensional systems, in particular polyhedral boranes. However, soon afterward, controversial difficulties arose with this approach. ... 

The acyclic polynomial of the Hiickel graph of the fullerene C24 ( >6d) (graph 42) is presented in Table 7. 

Nonisomorphic graphs with identical acyclic polynomials are called acyclic isospectral graphs. As an example, the pair of isospectral graphs representing l-methyl-2-propylcyclopro-pane, graph 43, and s-butylcyclopropane, graph 44, is presented ... 

From the definitions of the characteristic and acyclic polynomials one could hardly anticipate any connection between them, A connection is made by the Miilheim polynomial Mu(G, f, x) which continuously transforms Ac(G, x) into Ch(G, x) when the parameter f changes from zero to unity, Gutman and Polansky defined the Miilheim polynomial with the equation ... 

If G is the Hiickel graph of a conjugated hydrocarbon with N vertices, then the free term of the acyclic polynomial, i.e., the coefficient of x, Ac(G,0), is connected with KSC(G) ... 

The relevant polynomial corresponding to the reference structure is called the acyclic (77JA1692,77MI4) or reference (76JA2750,76JA6840) polynomial and since r(s) = 0, it has the form... 

Hosoya29) extended the Altenburg polynomial (originally devised for acyclic graphs) to cyclic graphs. 

Alternatively, for acyclic graphs Z can be defined as the sum of the absolute values of coefficients in the characteristic polynomial PH(G, x) ... 

For any acyclic graphs two general rules are observed i) the power of x decreases by two, and ii) the absolute values of T (G", x) coefficients are equal to the coefficients of the - Z-counting polynomial Q (G, x). Therefore, the characteristic polynomial for an acyclic graph is given by ... 

Therefore, the Hosoya Z index can be obtained from the polynomial for jc = 1. For acyclic graphs the Z-counting polynomial coefficients a(G, k) coincide with the absolute values of the coefficients of the graph - characteristic polynomial T [Nikolic etal, 1992]. 

It is noteworthy that 7z and 7z indices coincide with information indices on polynomial coefficients for acyclic graphs. 

In acyclic graphs, Z and Z are equal, the - Z-counting polynomial being in this case coincident with the characteristic polynomial. 

Note that, 2-methylpentane being an acyclic molecule, the detour polynomial coincides with the distance polynomial. 

By analogy with the Hosoya Z index that, for acyclic graphs, can be calculated as the sum of the absolute values of the coefficients of the characteristic polynomial of the adjacency matrix, the stability index (or modified Z index) is a molecular descriptor calculated for any graph as the sum of the absolute values of the coefficients C2i appearing alternatively in the characteristic polynomial of the adjacency matrix [Hosoya, Hosoi et al., 1975] ... 

Information indices on polynomial coefHcients are information indices defined as total information content and mean information content based on the partition of the coefficients of the characteristic polynomial of the graph. For acyclic molecules they coincide with the Hosoya total information index and Hosoya mean information index, respectively. 

For acyclic graphs, the matching polynomial coincides with the —> graph characteristic polynomial. Moreover, it was demonstrated the following relationship between the Z-counting and matching polynomials [Hosoya, 2003] ... 

If the summation in eq.(57) is performed only over those permutations P C P produced by independent transpositions, the acyclic or matching polynomial a( 7,A) of the graph Q is derived. The transpositions involved here have the cycle structure The acyclic... 

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