Iranian Journal of Mathematical Sciences and Informatics مجله علوم ریاضی و انفورماتیک IJMSI Basic Sciences http://ijmsi.ir 1 admin 1735-4463 2008-9473 8 10.52547/ijmsi 14 8888 13 en jalali 1398 7 1 gregorian 2019 10 1 14 2 online 1 fulltext
en On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs تخصصي Special پژوهشي Research paper <p>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)\$.</p> <p><br> 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|\$.<br> noindent<br> 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.}</p> Complementary distance matrix, Complementary distance signless Laplacian eigenvalues, Complementary distance signless Laplacian energy, Diameter. 105 125 http://ijmsi.ir/browse.php?a_code=A-10-2456-1&slc_lang=en&sid=1 H. S. Ramane hsramane@yahoo.com `10031947532846007808` 10031947532846007808 No Department of Mathematics, Karnatak University, Dahrwad- 580003, India. G. A. Gudodagi gouri.gudodagi@gmail.com `10031947532846007809` 10031947532846007809 No Department of Mathematics, KLE Societys, G. I. Bagewadi Arts, Science and Commerce College, Nipani 591237, Karnataka, India. V. V. Manjalapur vinu.m001@gmail.com `10031947532846007810` 10031947532846007810 Yes Department of Mathematics, KLE Societys, Basavaprabhu Kore Arts, Science and Commerce College, Chikodi 591201, Karnataka, India. A. Alhevaz a.alhevaz@shahroodut.ac.ir `10031947532846007811` 10031947532846007811 No Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box: 316-3619995161, Shahrood, Iran.