20 research outputs found
The Randic index and the diameter of graphs
The {\it Randi\'c index} of a graph is defined as the sum of
1/\sqrt{d_ud_v} over all edges of , where and are the
degrees of vertices and respectively. Let be the diameter of
when is connected. Aouchiche-Hansen-Zheng conjectured that among all
connected graphs on vertices the path achieves the minimum values
for both and . We prove this conjecture completely. In
fact, we prove a stronger theorem: If is a connected graph, then
, with equality if and only if is a path
with at least three vertices.Comment: 17 pages, accepted by Discrete Mathematic
The quotients between the (revised) Szeged index and Wiener index of graphs
Let and be the Szeged index, revised Szeged index and
Wiener index of a graph In this paper, the graphs with the fourth, fifth,
sixth and seventh largest Wiener indices among all unicyclic graphs of order
are characterized; as well the graphs with the first, second,
third, and fourth largest Wiener indices among all bicyclic graphs are
identified. Based on these results, further relation on the quotients between
the (revised) Szeged index and the Wiener index are studied. Sharp lower bound
on is determined for all connected graphs each of which contains
at least one non-complete block. As well the connected graph with the second
smallest value on is identified for containing at least one
cycle.Comment: 25 pages, 5 figure
Laplacian coefficients of unicyclic graphs with the number of leaves and girth
Let be a graph of order and let be the characteristic polynomial of its
Laplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c},
M. Ili\'{c}, Laplacian coefficients of trees with given number of leaves or
vertices of degree two, Linear Algebra and its Applications
431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian
coefficients in the set of all -vertex unicyclic graphs
with the number of leaves , we investigate properties of the minimal
elements in the partial set of the Laplacian
coefficients, where denote the set of -vertex
unicyclic graphs with the number of leaves and girth . These results are
used to disprove their conjecture. Moreover, the graphs with minimum
Laplacian-like energy in are also studied.Comment: 19 page, 4figure
A proof for a conjecture on the Randić index of graphs with diameter
AbstractThe Randić index R(G) of a graph G is defined by R(G)=∑uv1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n≥3 vertices with the Randić index R(G) and the diameter D(G), R(G)−D(G)≥2−n+12andR(G)D(G)≥n−3+222n−2, with equalities if and only if G is a path. In this work, we show that this conjecture is true for trees. Furthermore, we prove that for any connected graph on n≥3 vertices with the Randić index R(G) and the diameter D(G), R(G)−D(G)≥2−n+12, with equality if and only if G is a path