Miroslav Petrović

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.

On the second largest eigenvalue of line graphs

Let G be a connected claw-free graph on n vertices. Let ζ3(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if ζ3(G) ≥ n - 3 then G is traceable or else G is one of graphs Gn each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K3. Moreover, it is shown that for any integer k ≥ 4 there exists a positive integer V(k) such that if ζ3(G) ≥ n - k, n > V((k) and G is non-traceable, then G is a factor of a graph Gn. Consequently, the problem HAMILTONIAN PATH restricted to claw-free graphs G = (V, E) (which is known to be NP-complete) has linear time complexity O(|E|) provided that ζ3(G) ≥

Finite type graphs and some graph operations, II

\frac{5}{6}|V| - 3

Lower bounds of the Laplacian graph eigenvalues

. This contrasts sharply with known results on NP-completeness among dense graphs.

Graphs with a Small Number of Nonnegative Eigenvalues

Abstract In this paper we study a finite type property of graphs obtained by some n -ary operations on infinite graphs, continuing earlier work of A. Torgasev and the author.

Graphs with small numbers of independent edges

Abstract In this paper we prove that all positive eigenvalues of the Laplacian of an arbitrary simple graph have some positive lower bounds. For a fixed integer k ⩾ 1 we call a graph without isolated vertices k -minimal if its k th greatest Laplacian eigenvalue reaches this lower bound. We describe all 1-minimal and 2-minimal graphs and we prove that for every k ⩾ 3 the path P k +1 on k + 1 vertices is the unique k -minimal graph.

Terminal Wiener index

Abstract. In this paper, by means of computer checking, all simple graphs with at most two nonnegative eigenvalues, and all minimal simple graphs with exactly two (respectively, three) nonnegative eigenvalues are determined.

The high-energy band in the photoelectron spectrum of alkanes and its dependence on molecular structure

The graphs with exactly one, two or three independent edges are determined.