The signless Laplacian coefficients and the incidence energy of the graphs without even cycles

Abstract

Let Φn,m be the set of all connected graphs of order n with m edges without even cycles, where n≤m≤32(n−1). We consider the signless Laplacian coefficients and the incidence energy in Φn,m. Two new graph transformations are proposed here. Among Φn,m, it is obtained that there is exactly one minimal element with respect to the quasi-ordering according to their signless Laplacian coefficients. Furthermore, the graph with the minimal incidence energy is found among Φn,m.