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1.
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
2.
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., PI(G) = ∑
e∈E(G)[n
eu
(e|G) + n
ev
(e|G)], 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 phenylenes is given, and a simple relation is established between the
PI index of a phenylene and of the corresponding hexagonal squeeze. 相似文献
3.
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
4.
Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs 总被引:1,自引:0,他引:1
The second Zagreb index M
2(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which,
and C
n
have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively. 相似文献
5.
A (3,6)-fullerene is a plane cubic graph whose faces are only triangles and hexagons. A connected graph G with at least \(2n+2\) vertices is said to be n-extendable if it has n independent edges and every set of n independent edges extends to a perfect matching of G. A graph G is said to be bicritical if for every pair of distinct vertices u and v, \(G-u-v\) has a perfect matching. It is known that every (3,6)-fullerene is 1-extendable, but not 2-extendable. In this short paper, we show that a (3,6)-fullerene G is bicritical if and only if G has the connectivity 3 and is not isomorphic to one graph (2,4,2). As a surprising consequence we have that a (3,6)-fullerene is bicritical if and only if each hexagonal face is resonant. 相似文献
6.
7.
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
8.
The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs.
Mei Lu: Partially supported by NSFC (No. 10571105). 相似文献
9.
Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v
1,v
2,...,v
n
}. The 2-degree of v
i
, denoted by t
i
, is the sum of degrees of the vertices adjacent to
. Let σ
i
be the sum of the 2-degree of vertices adjacent to v
i
. In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, t
i
, and σ
i
, from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve
some known results for trees. 相似文献
10.
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
11.
The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. Let T be a tree with n vertices and k pendant vertices. In this paper, we give a sharp upper bound on Randić index of T. 相似文献
12.
Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e.,
, where d
G
(u, v) is the distance between vertices u and v in G. In this paper, we first give a new formula for calculating the Wiener index of an (n,n)-graph according its structure, and
then characterize the (n,n)-graphs with the first three smallest and largest Wiener indices by this formula. 相似文献
13.
The connectivity index χ1(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized.
Mei Lu is partially supported by NNSFC (No. 10571105). 相似文献
14.
15.
Denote by
n
the set of the hexagonal chains with n hexagons. For any B
n
n
, let m
k
(B
n
) and i
k
(B
n
) be the numbers of k-matchings and k-independent sets of B
n
, respectively. In the paper, we show that for any hexagonal chain B
n
n
and for any k0, m
k
(L
n
)m
k
(B
n
)m
k
(Z
n
) and i
k
(L
n
)i
k
(B
n
)i
k
(Z
n
), with left equalities holding for all k only if B
n
=L
n
, and the right equalities holding for all k only if B
n
=Z
n
, where L
n
and Z
n
are the linear chain and the zig-zag chain, respectively. These generalize some related results known before. 相似文献
16.
Let G be a simple graph with adjacency matrix A(G) and (G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Kleins idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811–823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results.1.If G is a bipartite graph containing no subgraph which is an even subdivision of K
2,3, then G has an orientation G
e such that (G,x) = det (xI-A(G
e
)), where A(G
e
) denotes the skew adjacency matrix of G
e.2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph
with 2n+2 vertices such that (G,x) is a factor of
.3.Let T be an arbitrary tree with n vertices. Then
, where
1
,
2
, ...,
n are the eigenvalues of T. 相似文献
17.
Denote by *
n
the set of all k
*-cycle resonant hexagonal chains with n hexagons. For any B
n
*
n
, let m(B
n
) and i(B
n
) be the numbers of matchings (=the Hosoya index) and the number of independent sets (=the Merrifield–Simmons index) of B
n
, respectively. In this paper, we give a characterization of the k
*-cycle resonant hexagonal chains, and show that for any B
n
*
n
, m(H
n
)m(B
n
) and i(H
n
)i(B
n
), where H
n
is the helicene chain. Moreover, equalities hold only if B
n
=H
n
. 相似文献
18.
Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d G (u, v) = 3}, where d G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given. 相似文献
19.
The Wiener index of a graph G is defined as
, where V(G) is the set of all vertices of G and for
denotes the length of a minimal path between x and y. A
C
4
C
8 net is a trivalent decoration made by alternating squares C
4 and octagons C
8. It can cover either a cylinder or a torus. In this paper, an algorithm for computing the distance matrix of a C
4
C
8(R) nanotorus T = T[p,q] is given. Using this matrix, the Wiener index of T is computed. 相似文献
20.
The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. We give a sharp lower bound on the Randić index of conjugated trees (trees with a perfect matching) in terms of the number
of vertices. A sharp lower bound on the Randić index of trees with a given size of matching is also given
Mei Lu: Partially supported by NNSFC (No. 60172005)
Lian-zhu Zhang: Partially supported by NNSFC (No. 10271105)
Feng Tian: Partially supported by NNSFC (No. 10431020) 相似文献