Chemistry-Relevant Isospectral Graphs. Acyclic Conjugated Polyenes

Hosoya, Haruo

doi:10.5562/cca3036

Cited by 5 publications

(5 citation statements)

References 24 publications

(30 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hosoya has found a number of examples of cospectral multiple coalescences (CMC) in his papers [21][22][23] using back-of-envelope calculations. Here we have instead developed an extensive suite of Java classes to exhaustively search for examples of multiple coalescences, based upon the existing Java framework for working with graphs [13], in which we have added an implementation of Samuelson-Berkowitz algorithm for computing characteristic polynomials [1].…”

Section: Computational Resultsmentioning

confidence: 99%

“…More recently, the well-known theoretical chemist Haruo Hosoya drew the attention in a series of papers [21][22][23] to a particular aspect of constructing cospectral graphs by using coalescence of graphs: besides attaching one copy of a rooted graph to either of two isospectral vertices, pairs of cospectral graphs can be obtained by attaching multiple copies of a rooted graph to different multisets of vertices in the underlying graph. Hosoya managed to find a number of pairs such as the one shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

“…In his works [21][22][23], Hosoya used the so-called Z-counting polynomial, which is equivalent to the characteristic polynomial in the case of trees, but not in the case of graphs that contain cycles. Hosoya studied cospectrality of coalescences in which two or three copies of a rooted graph are attached to the vertices of an underlying tree, and proposed a general expectation that the characteristic polynomial of the multiple coalescence depends on the family of characteristic polynomials of vertex-deleted subgraphs of the underlying tree T .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Hosoya's dormants and sprouts

Al-Yakoob¹,

Kanso²,

Stevanović³

2021

Preprint

View full text Add to dashboard Cite

The study of cospectral graphs is one of the traditional topics of spectral graph theory. Initial expectation by theoretical chemists Günthard and Primas in 1956 that molecular graphs are characterized by the multiset of eigenvalues of the adjacency matrix was quickly refuted by the existence of numerous examples of cospectral graphs in both chemical and mathematical literature. This work was further motivated by Fisher in 1966 in the influential study that investigated whether one can "hear" the shape of a (discrete) drum, which has led over the years to the construction of many cospectral graphs. These findings culminated in setting the ground for the Godsil-McKay local switching and the Schwenk's use of coalescences, both of which were used to show (around the 1980s) that almost all trees have cospectral mates. Recently, enumerations of cospectral graphs with up to 12 vertices by Haemers and Spence and by Brouwer and Spence have led to the conjecture that, on the contrary, "almost all graphs are likely to be determined by their spectrum". This conjecture paved the way for myriad of results showing that various special types of graphs are determined by their spectra.On the other hand, in a recent series of papers, Hosoya drew the attention to a particular aspect of constructing cospectral graphs by using coalescences: that cospectral graphs can be constructed by attaching multiple copies of a rooted graph in different ways to subsets of vertices of an underlying graph. The principal focus of this research effort is to address the expectations and questions raised in Hosoya's papers. We give an explicit formula for the characteristic polynomial of such multiple coalescences, from which we obtain a necessary and sufficient condition for cospectrality of these coalescences. We enumerate such pairs of cospectral multiple coalescences for a few families of underlying graphs, and show the infinitude of cospectral multiple coalescences having paths as underlying graphs, which were deemed rare by Hosoya.

show abstract

Section: Computational Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Hosoya's dormants and sprouts

Al-Yakoob¹,

Kanso²,

Stevanović³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Some attention has been directed to the occurrence of excessive (eigenvalue) degeneracies [17][18][19][20][21][22][23], sometimes viewed as manifestations of "local" structures [24][25][26][27][28][29][30][31][32]. Since degeneracies arose early on in the context of symmetry (and group theory), there has been a natural effort [33][34][35][36][37][38][39] to investigate excessive degeneracies in terms of symmetries, most simply in terms of a graph's automorphism group, which generally goes beyond standard point-group symmetries. The much studied [40][41][42][43][44][45] eigen-solution to the "Bethe tree" manifests high degeneracies.…”

Section: Preview and Frameworkmentioning

confidence: 99%

Eigen-Persistence in Graphs

Klein,

Seligman,

Sadurnı́

2024

MATCH

View full text Add to dashboard Cite

The idea of eigen-solution "persistence" from a suitable subgraph into a parent (molecular) graph G are formalized, in different ways for different cases. Most of it is based on the identification of suitably separated subgraphs sharing common eigenvalues, such that the subgraphs are all isomorphic. We recall Hall’s embedding method to identify adjacency-matrix eigen-solutions of a graph G as persistent from suitable disjoint subgraphs. General rigorous results are obtained for special embeddings, including cases where the subgraphs need not be isomorphic, but rather only share common eigenvalues. The question of accidental degeneracies is addressed, as well as the role of some sort of "local symmetries". Especially the mode of interconnection amongst a suitable family of subgraphs is addressed.

show abstract

“…By choosing every i,j pair, excluding the path i,j; repeating the above counting algorithm; and leaving the diagonal null, one would arrive at the Hosoya matrix, exemplified for G 1 [128]. Recently, Hosoya discovered new features of IS tree graphs from the analysis of isospectrality of conjugated polyenes [129,130].…”

Section: Hosoya Polynomialmentioning

confidence: 99%

Counting Polynomials in Chemistry: Past, Present, and Perspectives

Joița,

Tomescu,

Jäntschi

2023

Symmetry

View full text Add to dashboard Cite

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research.

show abstract

Chemistry-Relevant Isospectral Graphs. Acyclic Conjugated Polyenes

Cited by 5 publications

References 24 publications

On Hosoya's dormants and sprouts

On Hosoya's dormants and sprouts

Eigen-Persistence in Graphs

Counting Polynomials in Chemistry: Past, Present, and Perspectives

Contact Info

Product

Resources

About