Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition B of block size v. For blocks B,C of B adjacent in the quotient graph ΓB, let k be the number of vertices in B adjacent to at least one vertex in C. In this paper we classify all possibilities for (Γ,ΓB, G) in the case where k = v − 1 ≥ 2 and B(α) = B(β) for adjacent vertices α, β of Γ, where for a vertex of Γ, say γ ∈ B,...

Residue Number System (RNS) has computational advantages for large integer arithmetic. It provides the benefits of parallel, carry-free, and high-speed arithmetic in addition, subtraction, and multiplication. However, overflow detection, sign detection, magnitude detection, and division are time-consuming operations in RNS. The most interesting one of the above operations is division, and many ...

Symmetric graphs of diameter 2 with complete normal quotients Carmen Amarra University of Western Australia, Australia [email protected] A graph has diameter 2 if it is not a complete graph and if every pair of nonadjacent vertices is joined by a path of length 2. Our general problem is to examine the overall structure of graphs which are both arc-transitive and of diameter 2 using norm...

This work examines the problem of clique enumeration on a graph by exploiting its covers. The principle inclusion/exclusion is applied to determine number cliques size $r$ in union set $\mathcal{C} = \{c_1, \ldots, c_m\}$ $m$ cliques. leads deeper examination sets involved and an orbit partition, $\Gamma$, power $\mathcal{P}(\mathcal{N}_{m})$ $\mathcal{N}_{m} \{1, m\}$. Applied cliques, this pa...

With any G-symmetric graph admitting a nontrivial G-invariant partition B, we may associate a natural ‘‘cross-sectional’’ geometry, namely the 1-design DðBÞ 1⁄4 ðB; BðBÞ; IÞ in which IC for 2B and C 2 BðBÞ if and only if is adjacent to at least one vertex in C, where B2B and BðBÞ is the neighbourhood of B in the quotient graph B of with respect to B. In a vast number of cases, the dual 1-design...

Additive decompositions of a meromorphic function give rise to quotient representations of a particular form. We raise the question which quotient representations of a given function arise in this way. This question is answered by means of two characterizations via different terms. We pay particular attention to functions belonging to various subclasses of the Nevanlinna class of functions with...

A graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ E(G)} consists of n integers. Inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of G and K2 , and the disjoint union of G a...

Let $A$ and $B$ be two disjoint subsets of the vertex set $V$ of a graph $G$. The set $A$ is said to dominate $B$, denoted by $A rightarrow B$, if for every vertex $u in B$ there exists a vertex $v in A$ such that $uv in E(G)$. For any graph $G$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_p}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i rightarrow V_j$ or $V_j rightarrow...

We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpi...