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1.
Let λ1 (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let
be the set of trees with perfect matchings on 2m vertices, and
. Among the trees in
, we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest
eigenvalue when
. Furthermore, it is proved that, for two trees T
1 and T
2 in
(m≥ 4), if
and Δ (T
1) > Δ (T
2), then λ1 (T
1) > λ1 (T
2). 相似文献
2.
The Hosoya polynomial of a chemical graph G is , where d
G
(u, v) denotes the distance between vertices u and v. In this paper, we obtain analytical expressions for Hosoya polynomials of TUC4C8(S) nanotubes. Accordingly, the Wiener index, obtained by Diudea et al. (MATCH Commun. Math. Comput. Chem. 50, 133–144, (2004)), and the hyper-Wiener index are derived.
This work is supported by the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Grant No. LZULL200809). 相似文献
3.
We calculated the specific heat C
V
of ND4Cl under the prediction of an Ising model presented here. Our calculated C
V
agrees with the experimental C
P
data for the multicritical (P = 0.1 MPa) and second order (P = 150 MPa) phase regions in this crystalline system. This agreement
is particularly good above T
C
250 K) for the multicritical regime and below T
C
261 K) for the second order phase transition in ND4Cl. This shows that the observed behaviour of ND4Cl can be well described by an Ising model close to the phase transitions considered in this crystalline system. 相似文献
4.
Unicyclic graphs possessing Kekulé structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C
l
contained in the graph, respectively; r and j positive integers. It is mathematically verified that for
and l = 2r + 1 or
has the minimal energy in the graphs exclusive of
, where
is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C
4 and then by attaching n/2 − 3 paths of length 2 to one of the two vertices;
is a graph obtained by attaching one pendant edge and n/2 − 2 paths of length 2 to one vertex of C
3. In addition, we claim that for
has the minimal energy among all the graphs considered while for
has the minimal energy.
相似文献
5.
The Merrifield–Simmons index of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex
sets of G. By T(n,k) we denote the set of trees with n vertices and with k pendent vertices. In this paper, we investigate the Merrifield–Simmons index for a tree T in T(n,k). For all trees in T(n,k), we determined unique trees with the first and second largest Merrifield–Simmons index, respectively. 相似文献
6.
Extremal Polyomino Chains on k-matchings and k-independent Sets 总被引:3,自引:0,他引:3
Denote by the set of polyomino chains with n squares. For any , let m
k
(T
n
) and i
k
(T
n
) be the number of k-matchings and k-independent sets of T
n
, respectively. In this paper, we show that for any polyomino chain and any , and , with the left equalities holding for all k only if T
n
=L
n
, and the right equalities holding for all k only if T
n
=Z
n
, where L
n
and Z
n
are the linear chain and the zig-zag chain, respectively.
This work is supported by NNSFC (10371102). 相似文献
7.
Let G be a graph and d
v
denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals. 相似文献
8.
Ji-Ming Guo 《Journal of mathematical chemistry》2008,43(2):713-718
The energy E(G) of a graph G is defined as the sum of the absolute values of all the eigenvalues of the adjacency matrix of the graph G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. In this paper, we show that if G = (V
1, V
2; E) is a bipartite graph with edges and , then
and
must hold.
相似文献
9.
10.
Tomislav P. Živković 《Journal of mathematical chemistry》2006,39(1):151-176
Time-dependent properties of a state
that interacts with an infinite dimensional quantum system
containing several one-parameter eigenvalue bands
are considered. This is done by a new mathematical method that produces correct results, however strong the interaction between
the state
and the system
. It is shown that in the case of the weak interaction one obtains standard results that are usually obtained within the formalism
of the perturbation expansion method. In particular, if the eigenvalue E of the state
is embedded inside the range
of the unperturbed eigenvalues, time evolution of the state
that is initially prepared in the state
has typical exponential decay behavior. One also reproduces standard results concerning probabilities of the transition of
the state
at infinite time (t=∞) into various eigenvalue bands. However, if the interaction is strong, one finds much more complex and much more complicated
behavior. 相似文献
11.
Radosław Szmytkowski 《Journal of mathematical chemistry》2009,46(1):231-260
The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated
Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q
n
(z). The derivatives and , all with m > n, are also evaluated. 相似文献
12.
The Padmakar–Ivan (PI) index of a graph G is defined as PI , where for edge e=(u,v) are the number of edges of G lying closer to u than v, and is the number of edges of G lying closer to v than u and summation goes over all edges of G. The PI index is a Wiener–Szeged-like topological index developed very recently. In this paper, we describe a method of computing
PI index of benzenoid hydrocarbons (H) using orthogonal cuts. The method requires the finding of number of edges in the orthogonal
cuts in a benzenoid system (H) and the edge number of H – a task significantly simpler than the calculation of PI index directly
from its definition.
On the eve of 70th anniversary of both Prof. Padmakar V. Khadikar and his wife Mrs. Kusum Khadikar. 相似文献
13.
Tomislav P. Živković 《Journal of mathematical chemistry》2006,39(2):295-344
A new method for the exact solution of the interaction of an isolated state
with an infinite dimensional quantum system §∞ b
containing several one-parameter eigenvalue bands
is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction
between the state
and the system
. It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the
formalism of the perturbation expansion method. In particular, due to the interaction with the system
, eigenvalue E of the state
shifts to a new position. In addition, if this eigenvalue is embedded inside the range
of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state
has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much
more complicated behavior 相似文献
14.
H. Taşeli 《Journal of mathematical chemistry》2008,43(1):237-251
Special values of monic polynomials y
n
(s), with leading coefficients of unity, satisfying the equation of hypergeometric type
have been examined in its full generality by means of a unified approach, where σ(s) and τ(s) are at most quadratic and a linear polynomial in the complex variable s, respectively, both independent of n. It is shown, without actually determining the polynomials y
n
(s), that the use of particular solutions of a second order difference equation related to the derivatives y
n
(m)(z) is sufficient to deduce special values for some appropriate s = z points. Hence the special values of almost all polynomials and their derivatives can be generated by the universal formula
in which and are the discriminant and the roots of σ(s), respectively, and denote a parameter depending on the coefficients of the differential equation. Furthermore, the interrelations that arise
between and are also introduced. Finally, special values corresponding to the limiting and exceptional cases have been presented explicitly
for completeness.
相似文献
15.
Structural and thermal properties of the 20Li2O-80TeO2 glass were studied using X-ray diffraction analysis and differential scanning calorimetry techniques to understand and control
the crystallization process on this glass. The γ-TeO2, α-TeO2 and α-Li2Te2O5 phases were identified during the crystallization in this glass. Activation energies and Avrami exponent n were calculated
from non-isothermal measurements for glasses with different particle size. The mean values of Avrami exponent were obtained for glasses with 63–75 and 45–63 μm particle sizes such as , but glasses with particle size 38–45 μm and smaller than 38 g,m presented . 相似文献
16.
Vertex induced subgraphs of directed de Bruijn graphs with labels of fixed length k and over α letter alphabet are (α,k)-labelled. DNA graphs are (4,k)-labelled graphs. Pendavingh et al. proved that it is NP-hard to determine the smallest value α
k
(D) for which a directed graph D can be (α
k
(D),k)-labelled for any fixed . In this paper, we obtain the following formulas: and for cycle C
n
and path P
n
. Accordingly, we show that both cycles and paths are DNA graphs. Next we prove that rooted trees and self-adjoint digraphs
admit a (Δ,k)-labelling for some positive integer k and they are DNA graphs if and only if Δ ≤ 4, where Δ is the maximum number in all out-degrees and in-degrees of such digraphs. 相似文献
17.
A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. If d
v
denotes the degree of the vertex v, then the zeroth-order general Randić index
of the graph G is defined as
, where α is a real number. We characterize, for any α, the (n,n + 1)-graphs with the smallest and greatest zeroth-order general Randić index. 相似文献
18.
O. V. Frank-Kamenetskaya I. V. Rozhdestvenskaya A. G. Shtukenberg I. I. Bannova Yu. A. Skalkina 《Structural chemistry》2007,18(4):493-503
The crystal structures of three birefringent grossular-andradite natural garnets Ca3(Al,Fe)2(SiO4)3 were investigated using single-crystal X-ray diffraction data (MoKα, number of reflections measured 8065, 10619, 9213; R = 2.81, 2.74, 3.26%). According to the values of unit cell constants, inconsistent intensities of reflections and appearance
of additional (forbidden) reflections explored garnets have different symmetry: cubic, sp. gr.
(Fe/(Fe + Al) = 0.078, Δn = 0.0002); orthorhombic, sp. gr. Fddd (Fe/(Fe + Al) = 0.58, Δn = 0.0089); triclinic, sp. gr.
or I1 and pseudo-orthorhombic (Fe/(Fe + Al) = 0.23, Δn = 0.0066). Careful refinement of all crystal structures in space groups
, Fddd and
has confirmed the symmetry reduction detected on the diffraction patterns and shown that dissymmetrization of cubic garnets
connects with partial ordering of trivalent cations over Y-sites. Direct linear relationship between Fe-occupancy, an average
Y–O bond lengths and octahedral O–O edges has been revealed. Cluster models of dissymmetrization have been regarded. Evidence
for the “growth dissymmetrization” phenomena (kinetic phase transformations) as the reasons of the symmetry reduction of cubic
garnets has been discussed. The reasonable assumption that the garnets crystal structures described as orthorhombic are triclinic,
but the deviations from the orthorhombic symmetry so small, that cannot be manifested by of X-ray diffraction study has been
taken. 相似文献
19.
The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let
denote the set of trees on n vertices and diameter d,
. Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in
with minimal energy. In this article, the trees in
with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35 相似文献
20.
Let G be an unicycle graph and d
v
the degree of the vertex v. In this paper, we investigate the following topological indices for an unicycle graph ,
, where m ≥ 2 is an integer. All unicycle graphs with the largest values of the three topological indices are characterized.
This research is supported by the National Natural Science Foundation of China(10471037)and the Education Committee of Hunan
Province(02C210)(04B047). 相似文献