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1.
The general sum-connectivity index of a graph G is defined as χ
α
(G) = ∑edges (d
u
+ d
v
)
α
, where d
u
denotes the degree of vertex u in G and α is a real number. In this report, we determine the minimum and the second minimum values of the general sum-connectivity
indices of n-vertex unicyclic graphs for non-zero α ≥ −1, and characterize the corresponding extremal graphs. 相似文献
2.
The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. For a given positive
integer d with , we characterize the graphs with minimal energy in the class of unicyclic graphs with n vertices and a given diameter d.
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3.
The spread s(G) of a graph G is defined as s(G) = max
i,j
|λ
i
− λ
j
|, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U
*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U
*(n,k), and the graph with the largest spread in U(n,k).
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4.
The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya
index of a graph is defined as the total number of the matchings of the graph. In this paper, we order a kind of trees with
given number of pendent vertices with respect to Merrifield-Simmons indices and Hosoya indices. 相似文献
5.
Hongbo Hua 《Journal of mathematical chemistry》2008,43(2):831-844
The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total
number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S
n-l+1 with any vertex of C
l
. By we denote the graph obtained by identifying one pendent vertex of the path P
n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).
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6.
Kenneth C. Millett 《Journal of computational chemistry》1987,8(4):536-548
We describe a mathematical method that can be employed to define stereotopological indices of placements of certain graphs in space. These indices are applied to successfully distinguish between configurations in a chemically interesting family of knotted and/or linked four-valent oriented graphs in space. The methods are fundamentally algebraic and combinatorial in nature and are most readily understood in the context of calculations and the study of several key examples that are presented. 相似文献
7.
8.
Ying Liu 《Journal of mathematical chemistry》2010,47(1):1-10
A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Hou (J Math Chem 29:163–168,
2001) first considered the minimal energy for general unicyclic graphs. In this paper, we determine the unicyclic graphs with
the minimal energy in Unl{\mathcal {U}_n^l} and the unicyclic graphs with the first forth smallest energy in Un (n 3 13){\mathcal {U}_n\,(n\geq 13)} vertices. 相似文献
9.
Michitaka Furuya Masanori Takatou Shoichi Tsuchiya 《Journal of mathematical chemistry》2018,56(2):606-617
A fullerene graph is a 3-connected cubic plane graph whose all faces are bounded by 5- or 6-cycles. In this paper, we show that a matching M of a fullerene graph can be extended to a perfect matching if the following hold: (i) three faces around each vertex in \(\{x,y:xy\in M\}\) are bounded by 6-cycles and (ii) the edges in M lie at distance at least 13 pairwise. 相似文献
10.
The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n
eu
(e|G) and n
ev
(e|G) over all the edges e = uv of a connected graph G, i.e.,
, where n
eu
(e|G) is the number of edges of G lying closer to u than to v and n
ev
(e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows
of hexagonal of various lengths. 相似文献
11.
Tomislav Došlić 《Journal of mathematical chemistry》2008,43(2):647-657
Saturation number of a graph G is the minimum possible size of a maximal matching in G. We establish improved upper and lower bounds on the saturation number in fullerene graphs and discuss their sharpness and quality. 相似文献
12.
Teodor Silviu Balaban Petru A. Filip Ovidiu Ivanciuc 《Journal of mathematical chemistry》1992,11(1):79-105
A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).Dedicated to Professor Alexandru T. Balaban on the occasion of his 60th anniversary. 相似文献
13.
A computer program in Pascal is developed for computing the matching polynomials of graphs and lattices. This program is based on the recursive relation for matching polynomials outlined by Hosoya [Bull. Chem. Soc. Jpn., 44 , 2332 (1971)], Gutman and Hosoya [Theor. Chim. Acta, 48 , 279 (1978)], and others. The graph whose matching polynomial is of interest is reduced recursively until the graph reduces to several trees. The characteristic polynomial of a tree is the same as the matching polynomial. The characteristic polynomials of resulting trees are computed using the computer program based on Frame's method developed by Balasubramanian [Theor. Chim. Acta, 65 , 49 (1984)]; J. Comput. Chem., 5 , 387 (1984). The resulting polynomials are then assembled to compute the matching polynomial of the initial graph. The program is especially useful in generating the matching polynomials of graphs containing a large number of vertices. The matching polynomials thus generated are potentially useful in several applications such as lattice statistics (dimer covering problem), aromaticity, valence bond methods (enumeration of perfect matchings) in the calculation of grand canonical partition functions, in the computation of thermodynamic properties of saturated hydrocarbons, and in chemical documentation. 相似文献
14.
R.B. Mallion 《Chemical physics letters》1975,36(2):170-174
A recent theorem due to Waller is applied to the molecular graph of a typical conjugated system (naphthalene) in order to demonstrate the enumeration of spanning trees, on each of which a “ring current” calculation may be based. 相似文献
15.
Computational algorithms are described which provide for constructing the set of associated edge-weighted directed graphs such that the average of the characteristic polynomials of the edge-weighted graphs gives the matching polynomial of the parent graph. The weights were chosen to be unities or purely imaginary numbers so that the adjacency matrix is hermitian. The computer code developed earlier by one of the authors (K.B.) is generalized for complex hermitian matrices. Applications to bridged and spirographs, some lattices and all polycyclic graphs containing up to four cycles are considered. 相似文献
16.
17.
Konstantinova EV Vidyuk MV 《Journal of chemical information and computer sciences》2003,43(6):1860-1871
In this paper we consider 13 information and topological indices based on the distance in a molecular graph with respect to their discrimination power. The numerical results of discriminating tests on 3490528 trees up to 21 vertices are given. The indices of the highest sensitivity are listed on the set of 1528775 alkane trees. The discrimination powers of indices are also examined on the classes of 849285 hexagonal, 298382 square, and 295365 triangular simply connected animals. The first class of animals corresponds to the structural formulas of planar benzenoid hydrocarbons. The values of all indices were calculated for all classes of animals as well as for the united set of 1443032 animals. The inspection of the data indicates the great sensitivity of four information indices and one topological index. 相似文献
18.
19.
20.
Xiu-Mei Zhang Yu-Qin Sun Hua Wang Xiao-Dong Zhang 《Journal of mathematical chemistry》2018,56(2):568-582
The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence. 相似文献