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Letandbe two vertex disjoint graphs of ordersandrespectively, whereandLetbe a specified edge ofsuch thatis isomorphic toLetbe a subset of the edge set ofand letdenote the subgraph ofinduced byLetbe the graph obtained by taking one copy ofandvertex disjoint copies ofand then pasting the edgein theth copy ofwith the edgewhereThen the copies of the graphthat are pasted to the edgesare called as edge-pockets, and we sayis a graph withedge-pockets. In this article, we prove some results describing the Laplacian (resp. adjacency) spectrum ofusing the Laplacian (resp. adjacency) spectra ofandThe complete Laplacian (resp. adjacency) spectrum ofis also described in some particular cases. As an application, we show that these results enable us to construct infinitely many pairs of Laplacian (resp. adjacency) cospectral graphs. [ABSTRACT FROM AUTHOR]

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)= 1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p, l, q) is a graph obtained by joining two vertex-disjoint cycles Cp and Cq by a path of length l≥ 0. Let B* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in B* which are minimal configurations. [ABSTRACT FROM AUTHOR]