Vol. 29, No. 3, 1969

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Eigenvalues of the adjacency matrix of cubic lattice graphs

Renu Chakravarti Laskar

Vol. 29 (1969), No. 3, 623–629

A cubic lattice graph is defined to be a graph G, whose vertices are the ordered triplets on n symbols, such that two vertices are adjacent if and only if they have two coordinates in common. If n2(x) denotes the number of vertices y, which are at distance 2 from x and A(G) denotes the adjacency matrix of G, then G has the following properties: (P1) the number of vertices is n3. (P2) G is connected and regular. (P3) n2(x) = 3(n 1)2. (P4) the distinct eigenvalues of A(G) are 3, n 3, 2n 3, 3(n 1). It is shown here that if n > 7, any graph G (with no loops and multiple edges) having the properties (P1)–(P4) must be a cubic lattice graph. An alternative characterization of cubic lattice graphs has been given by the author (J. Comb. Theory, Vol. 3, No. 4, December 1967, 386–401).

Mathematical Subject Classification
Primary: 05.40
Received: 10 May 1968
Published: 1 June 1969
Renu Chakravarti Laskar