Mehdi Azadi Motlagh

Let $G$ be a graph with the vertex set $V(G)=\{v_1,\ldots,v_n\}$. The transmission of the vertex $v_i\in V(G)$, denoted by $\sigma_G(v_i)$, is defined to be the sum of distances between $v_i$ and any other vertices in $G$ , i.e., $\sigma_G(v_i)=\sum_{j=1}^n{{{d}_{G}}(v_i,v_j)}$. The signless Laplacian transmission matrix of $G$ is defined as $L_{Tr}^+(G)=\diag(\sigma_G(v_1),\cdots,\sigma_G(v_n))+A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $q_1^+,\ldots,q_n^+$ be the eigenvalues of $L_{Tr}^+(G)$. Then a transmission version of Laplacian energy of $G$ is defined as $EL_{Tr}^+(G)=\sum_{i=1}^n\Big|\mu_i^+ - \frac{2W(G)}{n}\Big|$. In this paper, we aim to obtain some bounds for $EL_{Tr}^+(G)$ in terms of other invariants of $G$ like, ordinary energy, Wiener and variable transmission Zagreb index.