Strong reciprocal eigenvalue property of a class of weighted graphs
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Let H be the class of connected bipartite graphs G   with a unique perfect matching 6c0ecf86f2afff192e94e" title="Click to view the MathML source">M. For 46d3fb7790bbd0ae685610fde090d7" title="Click to view the MathML source">G∈H, let WG be the set of weight functions w on the edge set E(G) such that w(e)=1 for each matching edge and w(e)>0 for each nonmatching edge. Let 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw denote the weighted graph with 46d3fb7790bbd0ae685610fde090d7" title="Click to view the MathML source">G∈H and w∈WG. The graph 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw is said to satisfy the reciprocal eigenvalue property, property (R)  , if 1/λ is an eigenvalue of the adjacency matrix A(Gw) whenever λ   is an eigenvalue of A(Gw). Moreover, if the multiplicities of the reciprocal eigenvalues are the same, we say 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has the strong reciprocal eigenvalue property, property (SR)  . Let 6c706137961bf352239bf60f0" title="Click to view the MathML source">Hg={G∈H|G/M is bipartite}, where G/M is the graph obtained from G   by contracting each edge in 6c0ecf86f2afff192e94e" title="Click to view the MathML source">M to a vertex.

Recently in [12], it was shown that if G∈Hg, then 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for some w∈WG if and only if 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for each w∈WG if and only if G is a corona graph (obtained from another graph H by adding a new pendant vertex to each vertex of H).

Now we have the following questions. Is there a graph G∈H∖Hg such that 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for each w∈WG? Are there graphs G∈H∖Hg such that 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw never has property (SR), not even for one w∈WG? Are there graphs 46d3fb7790bbd0ae685610fde090d7" title="Click to view the MathML source">G∈H such that 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for some w∈WG but not for all w∈WG? In this article, we supply answers to these three questions. We also supply a graph class larger than 6c3a95935bc9b2de89" title="Click to view the MathML source">Hg where for any graph G  , if 46" class="mathmlsrc">46.gif&_user=111111111&_pii=S0024379516304578&_rdoc=1&_issn=00243795&md5=09d4f46093fbdabd02d46a3d60da25e9" title="Click to view the MathML source">Gw has property (SR) for one w∈WG, then G is a corona graph.

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