There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons

https://www.math.ucdavis.edu/~saito/data/graphlap/merris-graphlap-eigvals.pdf

Laplacian matrices of graphs: a survey

A review of the aqueous electrochemical reduction of CO2 to hydrocarbons at copper

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.

Wiener Index of Trees: Theory and Applications

A topological index Z is proposed for a connected graph G representing the carbon skeleton of a saturated hydrocarbon. The integer Z is the sum of a set of the numbers p(G,k), which is the number of ways in which such k bonds are so chosen from G that no two of them are connected. For chain molecules Z is closely related to the characteristic polynomial derived from the topological matrix. It is found that Z is correlated well with the topological nature of the carbon skeleton, i.e., the mode of branching and ring closure. Some interesting relations are found, such as a graphical representation of the Fibonacci numbers and a composition principle for counting Z. Correlation of Z with boiling points of saturated hydrocarbons is pointed out.

Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons

On some counting polynomials in chemistry

A modified topological index
 
$$\tilde Z_G $$

 is proposed to be defined as
 
$$\tilde Z_G = \sum\limits_{k = 0}^{[N/2]} {( - 1)^k } a_{2k} $$

 for characterising theπ-electronic system of a conjugated hydrocarbonG withN carbon atoms, wherea
2k
 is the coefficient of the characteristic polynomial ofG defined as
 
$$P_G (X) = ( - 1)^N \det |A - XE| = \sum\limits_{k = 0}^N { a_k X^{N - k} } $$

 with an adjacency matrixA and the unit matrixE. 
$$\tilde Z_G $$

 is identical toZ

 G
 for a tree graph, or a chain hydrocarbon.Z

 G
 increases with a (4n+2)-membered ring formation and decreases with a 4n-membered ring formation. The totalπ-electron energyE

 π
 of the Huckel molecular orbital is shown to be related with
 
$$\tilde Z_G $$

 asE

 π
 =Cln
 
$$\tilde Z_G $$

 . With this relation generalised and extended Huckel rules for predicting the stability of an arbitrary network are proved.

A topological index for the totalπ-electron energy

Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons.

A general and simple method for graphical enumeration of the coefficients of the secular polynomials of the Huckel molecular orbitals is presented with examples. The essential procedure is to count for the graph and its subgraphs the non-adjacent numbers, p(G, k)'s, which appear as the coefficients of the Z-counting polynomial Q(Y). Two composition principles are proposed and shown to simplify these procedures to a great extent. Application to biphenylene shows the superiority of this method to others, e.g., the method of polygons.

Haruo Hosoya

Papers

Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons

On some counting polynomials in chemistry

A topological index for the totalπ-electron energy

Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons.

Graphical enumeration of the coefficients of the secular polynomials of the Hückel molecular orbitals