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Conceived and designed the experiments: MD LAJM AG. Performed the experiments: MD LAJM AG. Analyzed the data: MD LAJM AG. Contributed reagents/materials/analysis tools: MD LAJM AG. Wrote the paper: MD LAJM AG.

The authors have declared that no competing interests exist.

In this paper, we introduce a novel graph polynomial called the ‘information polynomial’ of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power.

The study of specific structural properties of graphs by using algebraic polynomial representations has been a well-known and fruitful concept for several decades

In this paper, we use the zeros of a novel graph polynomial to derive molecular descriptors. Before outlining the contribution of our paper, we briefly sketch some approaches surveyed by Randić et al.

The main contribution of our paper is as follows: Firstly, we define a novel graph polynomial which we call the

In this section, we define a novel graph polynomial as well as some spectral molecular descriptors by combining information-theoretic and algebraic methods

In particular, it was shown

We now start with a probability distribution

Let's introduce this framework formally. The quantities serving as vertex probabilities are defined by

We set

We remark that starting from Equation (8), (9), we see immediately that

In the following, we define some novel descriptors derived from the just introduced polynomial. The idea is to use the underlying zero distribution of

The aim of this section is to evaluate the just defined descriptors (see previous section) in terms of their uniqueness (degeneracy)

AG 3982: To create this database, we used the benchmark database called Ames mutagenicity

MS 2265: To create this dataset, the mass spectral database NIST

We implemented the novel descriptors and performed all involved calculations by using the programming language R

We now interpret the results we have obtained from calculating the sensitivity index

Descriptor |
||

0.999117 | 0.998995 | |

0.997351 | 0.994224 | |

0.998234 | 0.997489 | |

0.988521 | 0.976645 | |

0.067108 | 0.343046 | |

0.232546 | 0.402562 | |

0.859602 | 0.938724 | |

0.883885 | 0.947513 |

Descriptor |
||

0.928918 | 0.964591 | |

0.699338 | 0.834003 | |

0.000883 | 0.000883 | |

0.01557 | 0.01557 |

Now, we evaluate the discrimination power of the polynomial-based descriptors. Corresponding to the fact that the characteristic polynomial of a graph is degenerated

Also,

% |
% |
|||||

MS 2265 | −0.02 | 18.99 | 7798.00 | 21419.00 | 0.25 | 0.69 |

AG 3982 | −0.02 | 109.00 | 20539.00 | 56159.00 | 0.25 | 0.70 |

As conclusive remarks, the obtained results (in particular, see

In this paper, we introduced a novel graph polynomial and derived some descriptors by using the underlying zero distribution. We summarize the main findings of our paper and some future ideas as follows:

We started from the idea to use the probability distribution

So far, spectra of graphs have already been used to characterize chemical graphs

It is well-known that various topological descriptors have been successfully used for structure-oriented drug design

There is a considerable body of literature dealing with examining the zeros of graph polynomials

Deriving special estimations for the largest positive zero of

Deriving special bounds (leading to intervals containing the real zeros of

Exploring the location of the zeros for similar types of graph polynomials. Particularly, we want to derive special bounds for special graph classes.

As mentioned, the largest positive eigenvalue of trees has been used as a measures for branching

We thank Danail Bonchev, Thomas Stoll and Abbe Mowshowitz for fruitful discussions.

Also, we are grateful to Katja Hansen and Kurt Varmuza for calling our attention to the Ames mutagenicity database and providing the database MS 2265.