Families Of Regular Graphs Research Articles

Generalized spectral characterizations of regular graphs based on graph-vectors

A graph G is said to be determined by its generalized spectra (DGS for short) if for any graph H, H and G are cospectral with cospectral complements implies H is isomorphic to G. In Wang (2017) [14], the author gave a simple condition for a graph to be DGS. More precisely, let G be a n-vertex graph with adjacency matrix AG, and let W=[e,AGe,…,AGn−1e] be the walk-matrix of G, where e is the all-one vector. A theorem of Wang states that if 2−⌊n/2⌋det⁡W is odd and square-free, then G is DGS.However, the above theorem fails for regular graphs. In this paper, we first generalize the notion of DGS by introducing a new matrix associated with graph G, which is essentially a rank-one perturbation of the adjacency matrix of G and is obtained by introducing a graph-vectorξG associated with G. For a fixed graph-vector ξG, define Φ(G)=det⁡(λI−(AG+tξGξGT)). A graph G is determined byΦ(G) (Φ-DS for short) if for any graph H, Φ(G)=Φ(H) implies that H is isomorphic to G. It is not difficult to show that if ξG is the all-one vector, then G is Φ-DS if and only if it is DGS. Thus, the notion of Φ-DS generalizes that of DGS in a natural way. Moreover, we give a simple arithmetic condition for a large family of regular graphs to be Φ-DS, which is analogous to the aforementioned theorem. Numerical experiments are also presented to illustrate the effectiveness of our results.

On the hardness of the Balanced Connected Subgraph Problem for families of Regular Graphs

On the hardness of the Balanced Connected Subgraph Problem for families of Regular Graphs

On small world non-Sunada twins and cellular Voronoi diagrams

Special infinite families of regular graphs of unbounded degree and of bounded diameter (small world graphs) are considered. Two families of small world graphs Gi and Hi form a family of non-Sunada twins if Gi and Hi are isospectral of bounded diameter but groups Aut(Gi) and Aut(Hi) are nonisomorphic. We say that a family of non-Sunada twins is unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. If all Gi and Hi are edge-transitive we have a balanced family of small world non-Sunada twins. We say that a family of non-Sunada twins is strongly unbalanced if each Gi is edge-transitive but each Hi is edge-intransitive. We use term edge disbalanced for the family of non-Sunada twins such that all graphs Gi and Hi are edge-intransitive. We present explicit constructions of the above defined families. Two new families of distance-regular—but not distance-transitive—graphs will be introduced.

Fault-Tolerant Resolvability and Extremal Structures of Graphs

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

New infinite family of regular edge-isoperimetric graphs

We introduce a new infinite family of regular graphs admitting nested solutions in the edge-isoperimetric problem for all their Cartesian powers. The obtained results include as special cases most of previously known results in this area.

Small Regular Graphs of Girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$. A corrigendum was added to this paper on 21 June 2016.

On classes of regular graphs with constant metric dimension

In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn, a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in [I. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88: 43–57]. We prove that these classes of regular graphs have constant metric dimension. It is natural to ask for the characterization of regular graphs with constant metric dimension.

Pseudo and strongly pseudo 2-factor isomorphic regular graphs and digraphs

A graph G is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of G. In Abreu et al. (2008) [3] we proved that pseudo 2-factor isomorphic k-regular bipartite graphs exist only for k≤3. In this paper we generalize this result for regular graphs which are not necessarily bipartite. We also introduce strongly pseudo 2-factor isomorphic graphs and we prove that pseudo and strongly pseudo 2-factor isomorphic 2k-regular graphs and k-regular digraphs do not exist for k≥4. Moreover, we present constructions of infinite families of regular graphs in these classes. In particular we show that the family of Flower snarks is strongly pseudo 2-factor isomorphic but not 2-factor isomorphic and we conjecture that, together with the Petersen and the Blanuša2 graphs, they are the only cyclically 4-edge-connected snarks for which each 2-factor contains only cycles of odd length.

Formula omitted]-coloring of Kneser graphs

A b -coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b -coloring using k colors is the b -chromatic number b ( G ) of G . The b -spectrum S b ( G ) of a graph G is the set of positive integers k , χ ( G ) ≤ k ≤ b ( G ) , for which G has a b -coloring using k colors. A graph G is b -continuous if S b ( G ) = the closed interval [ χ ( G ) , b ( G ) ] . In this paper, we obtain an upper bound for the b -chromatic number of some families of Kneser graphs. In addition we establish that [ χ ( G ) , n + k + 1 ] ⊂ S b ( G ) for the Kneser graph G = K ( 2 n + k , n ) whenever 3 ≤ n ≤ k + 1 . We also establish the b -continuity of some families of regular graphs which include the family of odd graphs.

On symplectic graphs modulo [formula omitted

In this work, we study a family of regular graphs using the 2 ν × 2 ν symplectic group modulo p n , where p is a prime and n and ν are positive integers. We find that this graph is strongly regular only when ν = 1 . In addition, we define the symplectic graphs of a symplectic space V over a commutative ring R and show that it is vertex transitive and edge transitive when R has stable range one, which is the case for Z p n .

Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r ≥ 4 , every r -regular graph of odd order n ≤ 17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r -regular graphs of even order (including some Hamiltonian graphs) have VMTLs.

Effective equidistribution of eigenvalues of Hecke operators

In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S ( N , k ) of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre's theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator T p acting on S ( N , k ) of degree less than or equal to d. This allows us to determine an effectively computable constant B d such that if J 0 ( N ) is isogenous to a product of Q -simple abelian varieties of dimensions less than or equal to d, then N ⩽ B d . We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general “all-purpose” equidistribution theorem.

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of ( p m + 2 ) -regular graphs of girth five and order 2 p 2 m , where p ⩾ 5 is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an upper bound for the least number f ( k ) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f ( k ) , k ⩾ 16 .

On the domination number of the generalized Petersen graphs

Graph domination numbers and algorithms for finding them have been investigated for numerous classes of graphs, usually for graphs that have some kind of tree-like structure. By contrast, we study an infinite family of regular graphs, the generalized Petersen graphs G ( n ) . We give two procedures that between them produce both upper and lower bounds for the (ordinary) domination number of G ( n ) , and we conjecture that our upper bound ⌈ 3 n / 5 ⌉ is the exact domination number. To our knowledge this is one of the first classes of regular graphs for which such a procedure has been used to estimate the domination number.

Families of Regular Graphs in Regular Maps

The question of when a given graph can be the underlying graph of a regular map has roots a hundred years old and is currently the object of several threads of research. This paper outlines this topic briefly and proves that a product of graphs which have regular embeddings also has such an embedding. We then present constructions for members of three families: (1) circulant graphs, (2) wreath graphs W(k, n), whose vertices are ordered pairs (i, j), 0⩽i<k, 0⩽j<n, and whose edges are all possible (i, j)−(i+1, j′), and (3) depleted wreath DW(k, n), the subgraph of W(k, n) left by removing all edges in which j=j′. We open the question of multiplicity of occurrence and we list the underlying graphs of rotary maps with no more than 50 edges.

ALGORITHMS TO REALIZE AN ARBITRARY BPC PERMUTATION IN CHORDAL RING NETWORKS WITH FAILURES

A family of regular graphs of degree 3, called chordal rings is presented as a possible candidate for the implementation of a distributed system and for fault-tolerant architectures. The symmetry of graphs makes it possible to determine message routing by using a simple distributed algorithm. Arbitrary data permutations are generally accomplished by sorting. For certain classes of permutations, however, there exist algorithms that are more efficient than the best sorting algorithm. One such class is the Bit Permute Complement (BPC) class of permutations. In this paper, we first develop algorithms requiring two token storage registers in each node to realize an arbitrary BPC permutation. We next evaluate its ability to realize BPC permutations in networks of arbitrary size by estimating the number of required routing steps when a single fault is present and when not.

The growth rate of the harmonious chromatic number

AbstractIn this paper we investigate the growth rate of the harmonious chromatic number of a family of regular graphs, as a function of the number of vertices, when the valency or the diameter is fixed. We also look at families of graphs where the harmonious coloring is minimal.

Analysis of Chordal Ring Network

A family of regular graphs of degree 3, called Chordal Rings, is presented as a possible candidate for the implementation of a local network of message-connected (micro) computers. For a properly constructed graph in this family having n nodes the diameter, or maximum length message path, is shown to be of 0(n 1/2). The symmetry of the graphs makes it possible to determine message routing by using a simple distributed algorithm. The given algorithm is also potentially useful for the determination of alternate paths in the event of node or link failure.