Lower bounds for the Laplacian spectral radius of graphs

Let G = (V, E) be a graph of order nand with the Laplacian spectral radius lambda(G). For v(i) is an element of V, denote the set of all neighbors of viby Niand its number by d(i). The maximum degree of G is denoted by Delta(G). It is shown that if G is connected and Delta(G) < n - 1 then lambda(G) >= max {m(i)' + (1 + (m(i)' - 1)(2)/d(2,i) ) d(i)/m(i)' : v(i) is an element of V}, where m(i)'= Sigma(vivj is an element of E)(d(j)-vertical bar N-i boolean AND N-j vertical bar) d(i) and d(2,i) is the number of vertices at distance two from v(i). Also it is shown that lambda(G) >= max {(p(ij) + (1 - p(ij))(2)/p(ij) max{1, d(i) - 1}) x vertical bar N-i boolean OR N-j vertical bar : v(i)v(j) is an element of E, d(i) >= d(j)}, where p(ij)= e(N-i,N-j-N-i)/d(i)(vertical bar N-i boolean OR N-j vertical bar-d(i)), e(N-i, N-j-N-i) is the number of edges between N-i and N-j - N-i. (c) 2021 Elsevier Inc. All rights reserved.