Forcing, Freedom, & Uniqueness in Graph Theory & Chemistry

Abstract

Abstract. Harary's & Randić's ideas of "forcing" & "freedom" involve subsets of double bonds of Kekule structure such as to be unique to that Kekule structure. Such forcing sets are argued to be greatly generalizable to deal with various other coverings, and thence forcing seems to be fundamental, and of notable potential utility. Various forcing invariants associated to (molecular) graphs ensue, with illustrative (chemical) examples and some mathematical consequences being provided. A complementary "uniqueness" idea is noted, and the general characteristic of "derivativity" of "forcing" is established (as is relevant for QSPR fittings). Different ways in which different sorts of forcings arise in chemistry are briefly indicated.(doi: 10.5562/cca2000)

Study on the normalized Laplacian of a penta‐graphene with applications

“…In the physical community, the “resistance distance” can be interpreted as follows: we put a unit resistor on each edge of a network graph, and the effective resistance between vertices i and j on the graph is just the resistance distance between them, denoted by r ij . Indeed, this novel‐based function plays a key role in the field of graph theory, some interesting applications of which are used in chemistry . One of the well‐known resistance distance‐based parameters, the Kirchhoff index , is defined as Kf ( G ) = ∑ i < j r ij . Klein and Randić showed that Kf ( G ) ≤ W ( G ), and the equality holds if and only if G is a tree.…”

On the normalized Laplacian of Möbius phenylene chain and its applications

“…Given any pair of vertices u i and u j in G, let r ij denote the resistance distance between u i and u j . This parameter has many interesting applications in quantum chemistry, as seen in references [5][6][7][8][9], among others. In reference [10], it is indicated that even for moderately sized graphs, it is often quite involved to determine the resistance distance between vertices in an arbitrary graph.…”

“…Then we provide a complete description of the sum of the normalized Laplacian eigenvalues' reciprocals and the product of the normalized Laplacian eigenvalues. Based on these obtained results, we determine the multiplicative degree-Kirchhoff index and the number of spanning trees of HM 6,4 n . At last we show that the multiplicative degree-Kirchhoff index of HM 6,4 n is approximately 1 3 of its Gutman index.…”

“…Based on these obtained results, we determine the multiplicative degree-Kirchhoff index and the number of spanning trees of HM 6,4 n . At last we show that the multiplicative degree-Kirchhoff index of HM 6,4 n is approximately 1 3 of its Gutman index. Given a matrix M, let M[{i, j, …, k}] denote the submatrix obtained by deleting the ith, jth, …, kth rows and corresponding columns of M.…”

On normalized Laplacians, multiplicative degree‐Kirchhoff indices, and spanning trees of the linear [n]phenylenes and their dicyclobutadieno derivatives

“…This novel distance function, namely the resistance distance, on a graph was first proposed by Klein and Randi c. [11] For any pair of vertices u i and u j in G, let r ij be the resistance distance between u i and u j when the two poles of a battery to be connected to these two vertices. This novel parameter is in fact intrinsic to the graph, which has some nice interpretations and applications in quantum chemistry (see References [12][13][14][15][16] for details). In fact, to determine the resistance distance between vertices on arbitrary graphs is often quite intensive, even for moderately sized graphs; see References [17].…”