On the sum of signless Laplacian spectra of graphs (original) (raw)

2019, Carpathian Mathematical Publications

For a simple graph G(V,E)G(V,E)G(V,E) with nnn vertices, mmm edges, vertex set V(G)=v1,v2,dots,vnV(G)=\{v_1, v_2, \dots, v_n\}V(G)=v1,v2,dots,vn and edge set E(G)=e1,e2,dots,emE(G)=\{e_1, e_2,\dots, e_m\}E(G)=e1,e2,dots,em, the adjacency matrix A=(aij)A=(a_{ij})A=(aij) of GGG is a (0,1)(0, 1)(0,1)-square matrix of order nnn whose (i,j)(i,j)(i,j)-entry is equal to 1 if viv_ivi is adjacent to vjv_jvj and equal to 0, otherwise. Let D(G)=diag(d1,d2,dots,dn)D(G)={diag}(d_1, d_2, \dots, d_n)D(G)=diag(d1,d2,dots,dn) be the diagonal matrix associated to GGG, where di=deg(vi),d_i=\deg(v_i),di=deg(vi), for all iin1,2,dots,ni\in \{1,2,\dots,n\}iin1,2,dots,n. The matrices L(G)=D(G)−A(G)L(G)=D(G)-A(G)L(G)=D(G)−A(G) and Q(G)=D(G)+A(G)Q(G)=D(G)+A(G)Q(G)=D(G)+A(G) are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph GGG. If 0=munleqmun−1leqcdotsleqmu10=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_10=munleqmun−1leqcdotsleqmu1 are the Laplacian eigenvalues of GGG, Brouwer conjectured that the sum of kkk largest Laplacian eigenvalues Sk(G)S_{k}(G)Sk(G) satisfies Sk(G)=sumlimitsi=1kmuileqm+k+1choose2S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}Sk(G)=sumlimitsi=1kmuileqm+k+1choose2 and this conjecture...