Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

On the multiplicity of Laplacian eigenvalues and Fiedler partitions

In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get values and multiplicities of Laplacian matrix eigenvalues. We prove that a vertex set reduction on graphs with (m,k)-star subgraphs is feasible, keeping the same eigenvalues with reduced multiplicity. Moreover, some useful eigenvectors properties are derived up to a product with a suitable matrix. Finally, we relate these results with Fiedler spectral partitioning of the graph. The physical relevance of the results is shortly discussed

A study on determination of some graphs by Laplacian and signless Laplacian permanental polynomials

AbstractThe permanent of an n × n matrix [Formula: see text] is defined as [Formula: see text] where the sum is taken over all permutations σ of [Formula: see text] The permanental polynomial of M, denoted by [Formula: see text] is [Formula: see text] where In is the identity matrix of order n. Let G be a simple undirected graph on n vertices and its Laplacian and signless Laplacian matrices be L(G) and Q(G) respectively. The permanental polynomials [Formula: see text] and [Formula: see text] are called the Laplacian permanental polynomial and signless Laplacian permanental polynomial of G respectively. A graph G is said to be determined by its (signless) Laplacian permanental polynomial if all the graphs having the same (signless) Laplacian permanental polynomial with G are isomorphic to G. A graph G is said to be combinedly determined by its Laplacian and signless Laplacian permanental polynomials if all the graphs having (i) the same Laplacian permanental polynomial as [Formula: see text] and (ii) the same signless Laplacian permanental polynomial as [Formula: see text] are isomorphic to G. In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph [Formula: see text] (whose all r non-pendant vertices have the same degree n) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs [Formula: see text] (for [Formula: see text]), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials