Chemistry-Relevant Isospectral Graphs . Acyclic Conjugated Polyenes *

Abstract: By using the topological index Z and Z-counting polynomial proposed by the present author isospectral (IS) pairs of acyclic conjugated polyenes (C2nH2n+2) were studied. Besides the hitherto known smallest pair of n = 6, four and twenty seven pairs of n = 7 and 8, respectively, were first reported. Inspection of these results revealed several new features of IS tree graphs, i.e., appearance of two pairs of endospectral vertices in a tree graph and existence of several families of IS pair graphs whose Z-indices systematically grow up to infinity. Further several IS pairs were found to be closely related with each other topologically and called “intrinsic” IS pairs. Important role of the Z-index for analyzing the IS graphs is demonstrated.

Let us take our stance on the ground of chemistry but not on abstract mathematics.Although the very beginning of the problem of IS graphs is the discovery of the following pair of graphs, [1] , most graph theorists start their discussion from the following two pairs of IS graphs, [3,4] .It is difficult to find any chemical problem relevant to these pairs of graphs.
On the other hand, the question "Can one hear the shape of a drum?" cast by Kac from the standpoint of mathematical physics is a very important issue also in chemistry. [26]In a sense it is a problem arisen from non-tree graphs.
In this respect the IS pair of is introduced in many papers, but the present author could not find any chemically attractive discussion except for the one by Balasubramanian [18] who analyzes the relation between HMO and PPP.Then let us turn to acyclic conjugated polyene molecules to find more chemistry-relevant problem.
Although Godsil and McKay performed extensive computer-search for the IS graphs up to n = 14, [2] they , T published only two smallest pairs of "cospectral trees with a 1-factor," among which only the following pair of graphs, 1 and 2, were found to be useful for our discussion.Here double bonds are drawn in accordance with the chemical manner. .

2
This is the smallest pair of IS acyclic conjugated polyenes, C12H14, which unfortunately, however, have not yet been synthesized nor isolated.
In this study isospectrality of larger acyclic conjugated polyenes was checked up to C16H18, and more than thirty IS pairs were discovered!Further, three kinds of such genealogy trees of IS polyene pairs were found that grow up to infinity.

THEORY
It has been shown that the characteristic polynomial PG(x) of tree graph G composed of N vertices, defined by the adjacency (A) and unit (E) matrices, can be obtained from the set of non-adjacent number p(G,k) as [27,28] (G tree).
The p(G,k) is the number of ways for choosing k disjoint edges from graph G.In mathematical terminology p(G,k) is the number of k-matchings. [2]For any graph p(G,0) is defined to be unity, and p(G,1) is nothing else but the number of edges of G.The topological index Z (or Z-index) of graph G is defined as the total sum of p(G,k), For this purpose the Z-counting polynomial QG(x), proposed by the present author is useful. [27]In the case of IS 1 and 2, they have the common QG(x) as giving Z = 196 = 14 2 .
As a matter of fact, QG(x) is essentially equivalent to the later-proposed matching polynomial, [29][30][31] using our p(G,k) numbers, and for tree graphs MG(x) is identical to PG(x).Thus MG(x) will not be treated here any more.
In order to obtain QG(x) for a moderately large graph the following recursion relation is useful.
where G-l stands for the subgraph of G obtained by deleting an edge l, while GΘl is obtained by deleting l together with all the edges incident to l.In both the cases the QG(x)'s of all the isolated components arisen from this deleting process are to be multiplied.Let us explain this procedure by taking 1 as an example.See Figure 1, where l is chosen at the small arrow.Then we get the QG(x) which is identical to Eq. ( 4).The dotted circle demonstrates the process of getting QGΘl(x).Note that in deriving QGΘl(x) the Q function for a vertex, or P1, has been set to be unity.In the above process several other path graphs, Pn, can be seen, whose Z-values, or the sums of the coefficients of QG(x), form the famous Fibonacci numbers, 1, 2, 3, 5, 8, 13, 21, etc. [27,28] Next, try to obtain the Q function of 2, again by using the recursion relation (6).In this case, although any edge can be chosen for l, the wisest choice is found to be the small arrow as indicated below.Now the readers can realize that the same recursion formula can be obtained as in the case of 1.Let us call such a pair of IS graphs as 1 and 2 "intrinsic IS graphs", while other IS pairs of graphs may be called "fortuitous IS graphs".

RESULTS AND DISCUSSION
By using these techniques we could find four and twenty seven pairs of IS graphs, respectively, among the isomers of tetradecaheptaene and hexadecaoctaene, as shown in Tables 1 and 2.
Although it is not applied to this pair of graphs, the function of the pair of small arrows is the same as already explained above for 1 and 2.
It is rather strange that almost no IS pair of acyclic conjugated polyene has ever been discussed irrespective of the keen continuous interest of mathematical chemists for almost half a century.In Table 3 the number of isomers and IS pairs of acyclic conjugated polyenes C2nH2n+2 are given.Although systematic analysis for larger polyenes has not yet been tried, the number of IS pairs of octadecanonaenes (n = 9) is whimsically estimated to be around 200 independently of Schwenk's assertion. [25]y scrutinizing the Z-indices of the (re)discovered 31 pairs of IS graphs it was observed that there are so many square numbers.That is: 196 = 14 2 , 529 = 23 2 , 1156 = 34 2 , 1296 = 36 2 , and 1369 = 37 2 .Further, it is curious to find 14 + 23 = 37.Then after comparing the structures of these polyenes very interesting relation was found.
See Table 4, where a family of IS graphs is growing from the pair of 1 (A6) and 2 (B6) by stepwise addition of an edge at the pair of * and # vertices, respectively.Reversely, if a pair of edges of A6 (or B6) marked with * (or #) are deleted, one gets A5 (or B5).Note that the graphs A5 and B5 are equal, with Z5 = 81 = 9 2 .If this process is repeated to A5 (or B5), one gets a disjoint pair of P4 graphs yielding Z4 = 5 2 .
All the Zn's above n = 4 are found to be square numbers and obey the following recursive relation, yielding where τ is the golden ratio, (1 5) / 2 1.618 ... ,   which is deduced from the following relation, Although detailed analysis has not yet been completed, the interesting feature of the genealogies observed here may be due to the appearance of two pairs of endospectral vertices [17,33,34] in a tree graph.
In Table 5 is given another genealogy of IS pairs of tree graphs originating from (C7, D7) which already appeared in Table 1.In this case also by "tracing-back" one can obtain (C6, D6) both of which are the pair of P6 graphs yielding Z6 = 13 2 .The Z-indices of this family are also square numbers and obey the same recursion relations, (7) and (9).
Finally a very interesting family of IS pairs were found as shown in Table 6, which starts from (E8, F8) but the next members E9 and F9 are identical and radicals.Further, pairs of polyene and radical alternately appear as n increases.The fact that both of E7 and F7 are the pairs of Z = 19 graphs as well as the cases with (A4, B4) may suggest a plausible explanation of these three families of IS graphs whose Z-indices are all square numbers.
In Tables 4-6 we cannot find any family of IS pair graphs whose Z-indices are not square number, in spite of our expectation that the probability of finding IS pair graphs is rapidly increasing with the size of graphs.As the IS pairs in Tables 1 and 2 will play a key role for clarifying this problem, study in this direction is being in progress.
Here detailed accounts are not given, we have found a number of growing families of IS pairs of polyene radicals whose Z indices are not square numbers, which will be reported in other papers.Finally a serious question will be hurled to us: "Why are you playing with non-existing polyene isomers?" Yes, except for very limited cases actually almost no isomer above n > 4 in Table 3 has ever been isolated, and until recently "polyene chemistry" has been left barren as a desert amongst the flourishing organic chemistry, simply due to scarce isolation of conjugated polyenes except for several limited number of groups, such as carotene, linear polyene, and dendralene.However, thanks to the rapid development of organic synthesis many interesting polyenes and polyynes begin to be synthesized or isolated, [35,36] but not enough.At present the chemistry of acyclic polyenes is still uncultivated both experimentally and theoretically, although this area is in between the chemistry of aromatic and anti-aromatic compounds and thus ought to have a very important role in organic chemistry.
On the other hand, in the area of aromaticity and anti-aromaticity various properties of a vast number of hypothetical compounds have also been analyzed and discussed to establish and enlarge the real world of chemical compounds by the extensive use of chemical thinking and technique.This is the reason why the present author is now cultivating the polyene chemistry by reexamining the concept of cross-conjugation. [37]ble 4.A genealogy of pairs of IS graphs (An, Bn) of C2nH2n+2 isomers, which grow up from n = 6.Crucial vertices are marked with * and #, where edge-growing occurs.Z indices and their recursive formula are also given.A6 = 1 and B6 = 2. See text for more explanation.Table 3.The numbers of isomers [32] and IS pairs of acyclic conjugated polyenes C2nH2n+2.
instead of being bothered by the characteristic polynomial and its eigenvalues, we can only play with p(G,k)'s and Z.Let us explain a simple procedure for obtaining p(G,k) and Z by taking path graph P5, or the carbon atom skeleton of n-pentane or pentadienyl radical, as an example.By definition p(G,0) = 1, and p(G,1) = 4 for P5, while p(G,2) = 3 as enumerated from the following set of diagrams: , , and , yielding Z = 1 + 4 + 3 = 8 in this case.If the Z-indices of a given pair of isomeric graphs are different, one can safely assert that they are not IS.If they happen to have a common Z value, their isospectrality can be checked just by comparing the set of p(G,k)'s without recourse to their spectra (the set of eigenvalues).

Table 6 .
A genealogy of IS pair graphs (En, Fn) of C2nH2n+2 isomers, which grow up from n=8.Crucial vertices are marked with * and #, where edge-growing occurs.