New lower bounds for eigenvalues of a simple graph are derived. Upper and lower bounds for eigenvalues of bipartite graphs are presented in terms of traces and degree of vertices. Finally a non-trivial lower bound for the algebraic connectivity of a connected graph is given.

Let G be a connected graph of order n. The diameter of G is the maximum distance between any two vertices of G. In the paper, we will give some lower bounds for the algebraic connectivity and the Laplacian spectral radius of G in terms of the diameter of G. © 2006 Elsevier Inc. All rights reserved. AMS classification: 05C50; 15A18

Two new models, NO-JAG and NN-JAG in order of increasing computation power, are introduced as extensions to the conventional JAG model. A space lower bound of R(log2 n/ loglog 11) is proved for the problem of directed st-connectivity on a probabilistic NN-JAG and a space upper bound of O(1og n ) is proved for the problem of directed st-nonconnectivity on a nondeterministic NO-JAG. It as also sh...

In this paper, we show that every 4-connected maximal plane graph with m finite faces other than the octahedron can be drawn in the plane so that at least (m+3)/2 faces are acute triangles. Moreover, this bound is sharp. © 2005 Published by Elsevier B.V.

A planar 3-connected graph G is essentially 4-connected if for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. B. Jackson and N.C. Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that |V (C)| ≥ 2n+4 5 . For a cubic essentially 4-connected planar graph G, B. Grünbaum, J. Malkevitch , and C.-Q. ...

We prove that there exists a function a : N0 × R+ → N such that (i) If G is a 4-connected graph embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ε), then G can be covered by two cycles each of which has length at least (1− ε)n. We apply this to derive lower bounds for the length of a longest cycle in a graph G on any fixed surface. Specifically, there exist ...

It is shown that in a 4-connected maximal planar graph there is for any four vertices a, b, c and d, a cycle in the graph that contains the four vertices and visits them in the order a, b, c and d.

The Randić index of a graph G, written R(G), is the sum of 1 √ d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d < D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G) ≥ √ dD d+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and...

A graph G is almost 4-connected if it is simple, 3-connected, has at least five vertices, and V (G) cannot be partitioned into three sets A,B,C in such a way that |C| = 3, |A| ≥ 2, |B| ≥ 2, and no edge of G has one end in A and the other end in B. A graph K is a subdivision of a graph G if K is obtained from G by replacing its edges by internally disjoint nonzero length paths with the same ends...

The Randić index R(G) of a graph G is defined by R(G) = ∑ uv 1 √ d(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+1 2 and R(G...