Graph theory in mathematics and computer science is the study of graphs that are structures with pairwise connections between any objects. Here, the spectral graph theory and molecular dynamics simulation method are used to describe both morphological variation of ion aggregates in high salt solutions and ion effects on water hydrogen-bonding network structure. From the characteristic value analysis of the adjacency matrices that are graph theoretical representations of ion clusters, ion networks, and water H-bond structures, we obtained the ensemble average eigenvalue spectra revealing intricate connectivity and topology of ion aggregate structure that can be classified as either ion cluster or ion network. We further show that there is an isospectral relationship between the eigenvalue spectra of ion networks in high KSCN solutions and those of water H-bonding networks. This reveals the isomorphic relationship between water H-bond structure and ion-ion network structure in KSCN solution. On the other hand, the ion clusters formed in high NaCl solutions are shown to be graph-theoretically and morphologically different from the ion network structures in KSCN solutions. These observations support the bifurcation hypothesis on large ion aggregate growth mechanism via either ion cluster or ion network formation. We thus anticipate that the present spectral graph analyses of ion aggregate structures and their effects on water H-bonding network structures in high salt solutions can provide important information on the specific ion effects on water structures and possibly protein stability resulting from protein-water interactions.