Let G be a simple graph and let G denote its complement. We say that G is integral if its spectrum consists of integral values. In this work we establish a characterization of integral graphs which belong to the class αKa,a ∪ βKb,b, where mG denotes the m-fold union of the graph G. AMS Mathematics Subject Classification : 05C50.

It is well-known that each nonnegative integral flow of a directed graph can be decomposed into a sum of nonnegative graph circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that indecomposable flows of graphs are those graph circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than that of...

We prove that every nite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive.

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n− 1} and vertices a and b are adjacent if gcd(a− b, n) ∈ D, where D ⊆ {d : d | n...

A graph is called integral if all its eigenvalues are integers. The quest for integral graphs was initiated by F. Harary and A. J. Schwenk [9]. All such cubic graphs were obtained by D. Cvetković and F. C. Bussemaker [2, 1], and independently by A. J. Schwenk [10]. There are exactly thirteen cubic integral graphs. In fact, D. Cvetković [2] proved that the set of regular integral graphs of a fix...

An orientation of a graph is acyclic (totally cyclic) if and only if it is a “positive orientation” of a nowhere-zero integral tension (!ow). We unify the notions of tension and !ow and introduce the so-called tension-!ows so that every orientation of a graph is a positive orientation of a nowhere-zero integral tension-!ow. Furthermore, we introduce an (integral) tension-!ow polynomial, which g...

It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

Let G be a simple graph and let G denote its complement. We say that G is integral if its spectrum consists of integral values. Let Kxa,yb = Ka,a,...,a,b,b...,b be the complete m-partite graph with xa + yb vertices, where x and y are positive integers and m = x + y. In this work we consider integral graphs which belong to the class αKxa,yb for any α > 1 and a > b, where mG denotes the m-fold un...