On some graphs which satisfy reciprocal eigenvalue properties

Research paper by S.K.Pandaa, S.Patib

Indexed on: 09 Nov '17Published on: 01 Oct '17Published in: Linear Algebra and its Applications

Abstract

We consider only simple graphs. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. On the class Hg<math class="math"><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">H</mi></mrow><mrow is="true"><mi is="true">g</mi></mrow></msub>[/itex] of such graphs G the equivalence of the following statements are known.i)The reciprocal of the spectral radius of the adjacency matrix A(G)<math class="math"><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo>[/itex] is the least positive eigenvalue of the adjacency matrix.ii)The graph is isomorphic to its inverse, where the inverse of a graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G)<math class="math"><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo>[/itex] via a diagonal matrix of ±1s.iii)The graph has the reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A(G)<math class="math"><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo>[/itex] is also an eigenvalue of A(G)<math class="math"><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo>[/itex].iv)The graph has the strong reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A(G)<math class="math"><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo>[/itex] is also an eigenvalue of A(G)<math class="math"><mi is="true">A</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo>[/itex] and they both have the same multiplicities.v)The graph is a corona graph, that is, it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex. It is known that the statements iii) and iv) are not equivalent, if we allow nonbipartite graphs. However, the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings, where we relax the condition of bipartiteness of the contracted graph.