I n t r o d u c t i o n . In this paper, we s tudy the following quest ion: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to T a i t [6], who conjectured tha t any planar 3-connected 3-regular graph is hamiltonian. T u t t e [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tu t t e ' s exam...

Katerinis, P., Regular factors in regular graphs, Discrete Mathematics 113 (1993) 269-274. Let G be a k-regular, (k I)-edge-connected graph with an even number of vertices, and let m be an integer such that 1~ m s k 1. Then the graph obtained by removing any k m edges of G, has an m-factor. All graphs considered are finite. We shall allow graphs to contain multiple edges and we refer the reader...

Let L be a finite set associated with cliques of a distance-regular graph of order (s, t), with d-cliques of Johnson graphs and antipodal distance-regular graphs of diameter d, respectively. If we partially order L by the ordinary inclusion, three families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials. c © 2007 E...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

— In this note, we obtain a characterization of short normal sequences in a finite Abelian p-group, thus answering positively a conjecture of W. Gao for a variety of such groups. Our main result is deduced from a theorem of N. Alon, S. Friedland and G. Kalai, originally proved so as to study the existence of regular graphs in almost regular graphs. In the special case of elementary p-groups, Ga...

A signed adjacency matrix is a {−1, 0, 1}-matrix A obtained from the adjacency matrix A of a simple graph G by symmetrically replacing some of the 1’s of A by −1’s. Bilu and Linial have conjectured that if G is k-regular, then some A has spectral radius ρ(A) ≤ 2 √ k − 1. If their conjecture were true then, for each fixed k > 2, it would immediately guarantee the existence of infinite families o...