Extremal arithmetic-geometric spectral radius of unicyclic graphs

Spectral graph theory has been widely used in many fields, including network science, chemistry, physics, biology and sociology. Spectral extremal graph theory aims to minimize or maximize some graph invariants over a class of graphs based on graph matrix. The arithmetic-geometric matrix of a graph G , denoted by AG(G) , is a square matrix whose (i , j)-entry is d(i) +d(j)/2?d(i)d(j, i)f v(i) and v(j) are adjacent in G, and 0 otherwise, where di is the degree of vertex vi. The largest arithmetic-geometric eigenvalue is the AG spectral radius of G, denoted as theta(1)(G). In this work, we investigate the extremal values on arithmetic-geometric spectral radius of n-vertex unicyclic graphs and characterize the unicyclic graphs that achieve the extremes. We show that, for n-vertex unicyclic graph G , 2 = theta(1)(C-n) < theta(1)(G) < theta(1)(S-3(+)), where the lower bound is achieved by C-n and the upper bound is achieved by S-3(+) which is obtained by attaching n- 3 pendant vertices to some fixed vertex of C-3.