TY - JOUR T1 - On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs TT - JF - IJMSI JO - IJMSI VL - 14 IS - 2 UR - http://ijmsi.ir/article-1-1017-en.html Y1 - 2019 SP - 105 EP - 125 KW - Complementary distance matrix KW - Complementary distance signless Laplacian eigenvalues KW - Complementary distance signless Laplacian energy KW - Diameter. N2 - Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in V(G)}[1+D-d_G(u, v)]$. Let $CT(G)=diag[CT_G(v_1), CT_G(v_2), ldots, CT_G(v_n)]$. The complementary distance signless Laplacian matrix of $G$ is $CDL^+(G)=CT(G)+CD(G)$. If $rho_1, rho_2, ldots, rho_n$ are the eigenvalues of $CDL^+(G)$ then the complementary distance signless Laplacian energy of $G$ is defined as $E_{CDL^+}(G)=sum_{i=1}^{n}left| rho_i-frac{1}{n}sum_{j=1}^{n}CT_G(v_j)right|$. noindent In this paper we obtain the bounds for the largest eigenvalue of $CDL^+(G)$. Further we determine Nordhaus-Gaddum type results for the largest eigenvalue. In the sequel we establish the bounds for the complementary distance signless Laplacian energy.} M3 ER -