Let G be a finite simple graph of order n, maximum degree ∆, and minimum degree δ. A compact regularization of G is a ∆-regular graph H of which G is an induced subgraph: H is symmetric if every automorphism of G can be extended to an automorphism of H. The index |H : G| of a regularization H of G is the ratio |V (H)|/|V (G)|. Let mcr(G) denote the index of a minimum compact regularization of G...

A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; ...

A characterisation is given of edge-transitive Cayley graphs of valency 4 on odd number of vertices. The characterisation is then applied to solve several problems in the area of edge-transitive graphs: answering a question proposed by Xu (1998) regarding normal Cayley graphs; providing a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex-stabiliser; const...

Let Φ(x, y) ∈ C[x, y] be a symmetric polynomial of partial degree d. The graph G(Φ) is defined by taking C as set of vertices and the points of V(Φ(x, y)) as edges. We study the following problem: given a finite, connected, d-regular graph H , find the polynomials Φ(x, y) such that G(Φ) has some connected component isomorphic toH and, in this case, if G(Φ) has (almost) all components isomorphic...

Let G=(V,E) be a graph, a subset X of V is an interval of G whenever for a, b E X and xE V X , (a,x)EE (resp. (x,a)EE) if and only if (b,x)EE (resp. (x,b)EE). For instance, 0, {x}, where x E V, and V are intervals of G, called trivial intervals. A graph G is then said to be indecomposable when all of its intervals are trivial. In the opposite case, we will say that G is decomposable. We now int...

The existence of a connected 12-regular {K4, K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 i...

A graph G is called a prism fixer if γ(G×K2) = γ(G), where γ(G) denotes the domination number of G. A symmetric γ-set of G is a minimum dominating set D which admits a partition D = D1 ∪ D2 such that V (G)−N [Di] = Dj , i, j = 1, 2, i 6= j. It is known that G is a prism fixer if and only if G has a symmetric γ-set. Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discus...

The p-competition graph G of a digraph D is a graph on the same vertex set as D, with x; y] 2 E(G) if and only if jOut(x) \Out(y)j p in D. In this paper we focus on the case in which D is a symmetric digraph ((a; b) is an arc in D if and only if (b; a) is an arc in D). We relate the problem to 2-step graphs, squares, and a generalization of the neighborhood graph called the p-neighborhood graph...

In our previous works we had developed a framework for self-reconfiguration planning based on graph signature and graph edit-distance. The graph signature is a fast isomorphism test between different configurations and the graph edit-distance is a similarity metric. But the algorithm is not suitable for modules with symmetry. In this paper we improve the algorithm in order to deal with symmetri...

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank amon...