Pentacyclic And Hexacyclic Harmonic Graphs

Let G be a simple graph of order n. clearly, G has exactly one main eigenvalue if and if G is regular. It is a long-standing problem of D.Cvetkovic to characterize graphs with exactly k(k > 2) main eigenvalues. Recently, A.Dress and Gutman gave an concept of harmonic graphs: The graph G is said to be harmonic if there exists a constant A, such that the equality A(G)d(G)=Ad(G) holds. One non-regular graph G isA-harmonic if only if G has exactly main eigenvalues A and zero. Earlier all harmonic trees were determined and all unicyclic , acyclic, bicyclic, tricyclic and tetracyclic harmonic graphs were characterized.In my paper, we go a step further and find all pentacyclic and hexacyclic harmonic graphs: (1) : There are exactly 62 connected pentacyclic harmonic graphs, where there are exactly 56 non-regular and 5 regular connected pentacyclic 3-harmonic graphs and there exist exactly 1 non-regular connected pentacyclic 4-harmonic graphs. (2) : there are exactly 77 connected hexacyclic harmonic graphs, where there are exactly 55 non-regular and 19 regular connected hexacyclic 3-harmonic graphs and there exist exactly 2 non-regular and 1 regular connected hexacyclic 4-harmonic graphs.