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be the degree of thevertex i.LetA(G) be the adjacency matrix of Gand let D(G)bethediagonal matrix of vertex degrees.The Laplacian matrix of Gis L(G)=D(G) −A(G).Clearly, L(G) is a real symmetric matrix. From this factand Gerˇsgorin’s theorem, it follows that its eigenvalues are nonnegative realnumbers. Moreover, since its rows sum to 0,0 is the smallest eigenvalueof L(G).In [7], some of the many results known for Laplacian matricesare given. Fiedler [3] proved that Gis a connected graph if and only if thesecond smallest eigenvalue of L(G) is positive. This eigenvalue is called thealgebraic connectivity of Gand it is denoted by a(G).This concept hasbeen studied by many authors. In section 3 of [7], some results concerninga(G) and some of its many applications are presented.Denote by B

THE SPECTRUM OF THE LAPLACIAN MATRIX OF A BALANCED 2p-ARY TREE