Research On Some Invariants Of Some Types Of Graphs

The first and the second Zagreb indices, adjacency spectral radius and signless Laplacian spectral radius are the invariants of graphs. This paper which stands on the basis of previous results, do further research on sharp bounds of Zagreb indices of unicyclic graphs, and compare the Zagreb indices for maximal outerplanar graphs; moreover, do further research on the issue of the signless Laplacian spectral radii of unicyclic graphs with matching number q, and of the spectral radii of n-vertex cacti with matching number q, respectively.●In the first two sections, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound discus-sion on the status quo, it fully shows the main work's necessity and innovation.●In Section 3, we study the sharp upper and lower bounds for the Zagreb indices among the n-vertex maximal outerplanar graphs, and among n-vertex outer-planar graphs (resp. maximal outerplanar graphs) with perfect matchings. And the corresponding graphs are characterized.●In Section 4, we compare the Zagreb indices for maximal outerplanar graphs.●In Section 5, we determine the first three largest signless Laplacian spectral radii together with the corresponding graphs among unicyclic graphs with n vertices and matching number q. And the corresponding graphs are charac-terized.●In Section 6, we determine the first and second largest spectral radii together with the corresponding graphs among cacti with n vertices and matching num-ber q. And the corresponding graphs are characterized.