Results 11  20
of
39
THE MINIMUM SPECTRAL RADIUS OF GRAPHS WITH A GIVEN CLIQUE NUMBER
, 2008
"... In this paper, it is shown that among connected graphs with maximum clique size ω, the minimum value of the spectral radius of adjacency matrix is attained for a kite graph P Kn−ω,ω, which consists of a complete graph Kω to a vertex of which a path Pn−ω is attached. For any fixed ω, a small interv ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
In this paper, it is shown that among connected graphs with maximum clique size ω, the minimum value of the spectral radius of adjacency matrix is attained for a kite graph P Kn−ω,ω, which consists of a complete graph Kω to a vertex of which a path Pn−ω is attached. For any fixed ω, a small interval to which the spectral radii of kites P Km,ω, m ≥ 1, belong is exhibited.
On Mean Distance and Girth
, 2008
"... We bound the mean distance in a connected graph which is not a tree in function of its order n and its girth g. On one hand, we show that mean distance is at most n+1 3 − g(g2 −4) 12n(n−1) if g is even and at most n+1 3 − g(g2 −1) 12n(n−1) if g is odd. On the other hand, we prove that mean distance ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We bound the mean distance in a connected graph which is not a tree in function of its order n and its girth g. On one hand, we show that mean distance is at most n+1 3 − g(g2 −4) 12n(n−1) if g is even and at most n+1 3 − g(g2 −1) 12n(n−1) if g is odd. On the other hand, we prove that mean distance is at least unless G is an odd cycle.
Solutions to two unsolved questions on the best upper bound for the Randić index R−1 of trees
 MATCH Commun. Math. Comput. Chem
"... The general Randic index R(G) of a graph G is dened as the sum of the weights (d(u)d(v)) of all edges uv of G, where d(u) denotes the degree of a vertex u in G and is an arbitrary real number. Clark and Moon gave the lower and upper bounds for the Randic index R1 of trees with order n. The lower b ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The general Randic index R(G) of a graph G is dened as the sum of the weights (d(u)d(v)) of all edges uv of G, where d(u) denotes the degree of a vertex u in G and is an arbitrary real number. Clark and Moon gave the lower and upper bounds for the Randic index R1 of trees with order n. The lower bound is sharp. However, a sharp upper bound has not been obtained for a long time. Clark and Moon proposed two unsolved questions on the upper bound. In this paper, we give positive answers to the two questions. We show that limn!1f(n)=n = 15=56, and give a sharp upper bound for which there are innitely many trees of order n whose values of Randic index R1 attain the bound. Supported by National Science Foundation of China. 1
An Updated Survey of Research in Automated Mathematical ConjectureMaking
"... This is an updated version of [33]. Research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described. The fundamental principle underlyin ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This is an updated version of [33]. Research on automated mathematical conjecturemaking is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful  and little known  program is comprehensively described. The fundamental principle underlying this program can be simply stated: make the strongest conjecture for which no counterexample is known. Conjecturemaking may be key to building machines with a wide variety of intelligent behaviors. If so, this principle should prove exceptionally useful.
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
On the Randic ́ index of cacti
"... The Randic ́ 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. In the paper, we give a sharp lower bound on the Randić index of cacti. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The Randic ́ 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. In the paper, we give a sharp lower bound on the Randić index of cacti.
HE(G) =
, 2009
"... Let G be a graph on n vertices with r: = ⌊n/2 ⌋ and let λ1 ≥ · · · ≥ λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as 2 ..."
Abstract
 Add to MetaCart
Let G be a graph on n vertices with r: = ⌊n/2 ⌋ and let λ1 ≥ · · · ≥ λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as 2
Bounds for the Hückel energy of a graph
"... Let G be a graph on n vertices with r: = ⌊n/2 ⌋ and let λ1 � · · · � λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as r∑ 2 λi, if n = 2r; i=1 HE(G) = r∑ 2 λi + λr+1, if n = 2r + 1. i=1 The concept of Hückel energy was introduced by Coulson as it gives a good ap ..."
Abstract
 Add to MetaCart
Let G be a graph on n vertices with r: = ⌊n/2 ⌋ and let λ1 � · · · � λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as r∑ 2 λi, if n = 2r; i=1 HE(G) = r∑ 2 λi + λr+1, if n = 2r + 1. i=1 The concept of Hückel energy was introduced by Coulson as it gives a good approximation for the πelectron energy of molecular graphs. We obtain two upper bounds and a lower bound for HE(G). When n is even, it is shown that equality holds in both upper bounds if and only if G is a strongly regular graph with parameters (n,k,λ,µ) = (4t 2 + 4t + 2, 2t 2 + 3t + 1, t 2 + 2t, t 2 + 2t + 1), for positive integer t. Furthermore, we will give an infinite family of these strongly regular graph whose construction was communicated by Willem Haemers to us. He attributes the construction to J.J. Seidel.