Let G be a generalized dicyclic group with identity 1. An inverse closed subset S of G∖{1} is called minimal if 〈S〉=G and there exists some s∈S such that 〈S∖{s,s−1}〉≠G. In this paper, we characterize distance-regular Cayley graphs Cay(G,S) under the condition minimal.

We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r−o(r) edge-disjoint Hamilton cycles. Here G is a robust outexpander if for every set S which is not too small and not too large, the ‘robust’ outneighbourhood of S is a little larger than S. T...

A Cayley (di)graph Cay(G,S) of a group G is called normal if the right regular representation in full automorphism Cay(G,S), and CI-(di)graph for every Cay(G,T),Cay(G,S)≅Cay(G,T) implies that there σ∈Aut(G) such Sσ=T. We call an NDCI-group or NCI-group all digraphs graphs are CI-digraphs CI-graphs, respectively. prove cyclic order n only 8∤n, either = 8 8∤n.

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound of O(n 2) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of s log n) for the expected cover time...

Let G be a graph of order n and let μ be an eigenvalue of multiplicity m. A star complement for μ in G is an induced subgraph of G of order n−m with no eigenvalue μ. In this paper, we study the maximal graphs as well as regular graphs which have Kr,s + tK1 as a star complement for eigenvalue 1. It turns out that some well known strongly regular graphs are uniquely determined by such a star comp...

We investigate the complexity of the Maximum Induced Matching problem on regular graphs and trees. The problem is shown to be NP-complete on regular graphs of degree 4s for any positive integer s. We also looks at very simple approximation algorithms: we show that the largest induced matching in a regular graph of degree d can be approximated with a performance ratio less than d. However a simp...

In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles h...

In this lecture, we will focus on expander graphs (also called expanders), which are pseudoran-dom objects in a more restricted sense than what we saw in the last two lectures. The reader is also referred to the monograph [1] and the tutorial slides [2] for more detailed surveys of today's topics. Expander graphs are universally useful in computer science and have many applications in de-random...