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A split graph is a graph whose vertices can be partitioned into a clique and an independent set. Most results in spectral graph theory do not address multigraph concerns. Exceptions are [2] and [4], but these papers present results involving a special class of underlying split graphs, threshold graphs, in which all pairs of nodes exhibit neighborhood nesting, and all multiple edges are confined to the clique.
We present formulas for the eigenvalues of some infinite families of regular split multigraphs in which all multiple edges occur between the clique nodes and cone nodes, with multiplicity of multiple edges \( \mu > 1 \) fixed, and which have integer eigenvalues for the adjacency, Laplacian, and signless Laplacian matrices.