First, regarding the mathematics of the middle-eigenvalue problem there are results for the case of a graph G with just a single perfect matching, especially if also the graph is bipartite.  See D. J. Klein & A. Misra, Croat. Chim. Acta 77 (2002) 179-191, where a transform (called "Kekulean") from an original G with a single perfect matching is made to another graph G' with signed edge weights such that the adjacency matrix A(G') is the inverse of the original A(G). Circumstances are identified where the signs on G' may be eliminated to leave an ordinary unsigned graph G" still with eigenvalues inverse to those of G. Further circumstances are found where G & G" are isomorphic.  Techniques for dealing with maximum eigenvalues (of A(G') or A(G")) are used to give information on the "middle" eigenvalues (of A(G)).
 
   Second, some comments might be made on the chemical context.  The adjacency-matrix eigenvalues nearest 0 are much considered in chemistry, as they locate the electrons most easily excited and the eigenvalue difference for middle eigenvalues gives an appropriate measure of the energy needed for excitation. In more detail, the eigenvalues of the adjacency matrix provide crude (Huckel-theoretic) estimates of 1-electron energies for the pi-electrons of conjugated carbon networks -- the other electrons being for the most part more tightly bound.  A full N-electron energy is then just a sum over 1-electron energies each multiplied by an occupation number n(e) for that eigenvalue e -- the occupation numbers taking values 0, 1, or 2 and summing to N.  For an electrically neutral conjugated-carbon network (as is a common circumstance), the total number N of such electrons matches the number of sites.  Then the most favored N-electron state for N=even has the N/2 largest eigenvalues e each with n(e)=2 and all other eigenvalues e' with n(e')=0.  For odd N=2k+1, the k largest e have n(e)=2, the next lowest eigenvalue e' has n(e')=1, and the k remaining lower eigenvalues e" have n(e")=0.  The gap of interest is the least energy difference between a level not doubly occupied and another not empty.  [A point of potential confusion is that the 1-electron eigenvalues are proportional to the eigenvalues with a proportionality which is negative; then the favorable ("ground-state") circumstance of occupying the largest eigenvalues corresponds to occupying the lowest energy 1-electron levels.]  From all this commentary it can be seen that for bipartite graphs the "middle" eigenvalues of interest are those nearest 0, while for nonbipartite graphs this is not necessarily so.  The chemical graphs of interest for conjugated-carbon networks have vertices just of degrees 1, 2, or 3.