Tetracyclic harmonic graphs

B. BoroviŠanin, I. Gutman, M. PetroviŠ

A graph $G$ on $n$ vertices $v_1,v_2,\ldots,v_n$ is said to be harmonic if $(d(v_1),d(v_2),\ldots,d(v_n))^t$ is an eigenvector of its $(0,1)$-a­acency matrix, where $d(v_i)$ is the degree (= number of first neighbors) of the vertex $v_i \ , \ i=1,2,\ldots,n$ . Earlier all acyclic, unicyclic, bicyclic and tricyclic harmonic graphs were characterized. We now show that there are 2 regular and 18 non-regular connected tetracyclic harmonic graphs and determine their structures.