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Bicyclic graphs with exactly two main signless Laplacian eigenvalues  [PDF]
He Huang,Hanyuan Deng
Mathematics , 2011,
Abstract: A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.
On signless Laplacian coefficients of unicyclic graphs with given matching number  [PDF]
Jie Zhang,Xiao-Dong Zhang
Mathematics , 2012,
Abstract: Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes all n-vertex unicyclic graphs with matching number $m$. We characterize the graphs which minimize all the signless Laplacian coefficients in the set $\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremal graphs which have minimal signless Laplacian coefficients in the set $\mathcal{G}(n)$ of all $n$-vertex unicyclic graphs with odd (resp. even) girth.
Maxima of the signless Laplacian spectral radius for planar graphs  [PDF]
Guanglong Yu
Mathematics , 2014,
Abstract: The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph $K_{2}\nabla P_{n-2}$ has the maximal signless Laplacian spectral radius among all planar graphs of order $n\geq 456$.
Laplacian and signless Laplacian spectral radii of graphs with fixed domination number  [PDF]
Rundan Xing,Bo Zhou
Mathematics , 2013,
Abstract: In this paper, we determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs respectively.
The signless Laplacian Estrada index of tricyclic graphs  [PDF]
Ramin Nasiri,Hamid Reza Ellahi,Gholam Hossein Fath-Tabar,Ahmad Gholami,Tomislav Do?li?
Mathematics , 2014,
Abstract: The signless Laplacian Estrada index of a graph $G$ is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$ where $q_1, q_2, \ldots, q_n$ are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we show that there are exactly two tricyclic graphs with the maximal signless Laplacian Estrada index.
The Signless Laplacian Estrada Index of Unicyclic Graphs  [PDF]
Hamid Reza Ellahi,Ramin Nasiri,Gholam Hossein Fath-Tabar,Ahmad Gholami
Mathematics , 2014,
Abstract: For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.
Edge Bipartiteness and Signless Laplacian Spread of Graphs  [PDF]
Yi-Zheng Fan,Shaun Fallat
Applicable Analysis and Discrete Mathematics , 2012, DOI: 10.2298/aadm120127003f
Abstract: Let $G$ be a connected graph, and let $eb(G)$ and $sq(G)$ be the edge bipartiteness and the signless Laplacian spread of $G,$ respectively. We establish some important relationships between $eb(G)$ and $sq(G),$ and prove $ sq(G) ge 2 Big(1+cos fracc{pi}{n} Big),$ with equality if and only if $G=P_n$ or $G=C_n$ in case of odd $n.$ In addition, we show that if $G eq P_n$ or $G eq C_{2k+1},$ then $sq(G) ge 4,$ with equality if and only if $G$ is one of the following graphs: $K_{1,3},$ $K_4,$ two triangles connected by an edge, and $C_n$ for even $n.$ As a consequence, we prove a conjecture of extsc{Cvetkovi'c, Rowlinson} and extsc{Simi'c} on minimal signless Laplacian spread [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd), {f81} (95) (2007), 11--27].
TOWARDS A SPECTRAL THEORY OF GRAPHS BASED ON THE SIGNLESS LAPLACIAN, III
Drago? Cvetkovi?,Slobodan K. Simi?
Applicable Analysis and Discrete Mathematics , 2010, DOI: 10.2298/aadm1000001c
Abstract: This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results.
On Maximum Signless Laplacian Estrada Indices of Graphs with Given Parameters  [PDF]
H. R. Ellahi,R. Nasiri,G. H. Fath-Tabar,A. Gholami
Mathematics , 2014,
Abstract: Signless Laplacian Estrada index of a graph $G$, defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$, where $q_1, q_2, \cdots, q_n$ are the eigenvalues of the matrix $\mathbf{Q}(G)=\mathbf{D}(G)+\mathbf{A}(G)$. We determine the unique graphs with maximum signless Laplacian Estrada indices among the set of graphs with given number of cut edges, pendent vertices, (vertex) connectivity and edge connectivity.
The spectra and the signless Laplacian spectra of graphs with pockets  [PDF]
Shu-Yu Cui,Gui-Xian Tian
Mathematics , 2014,
Abstract: Let $G[F,V_k,H_v]$ be the graph with $k$ pockets, where $F$ is a simple graph of order $n\geq1$, $V_k=\{v_1,\ldots,v_k\}$ is a subset of the vertex set of $F$ and $H_v$ is a simple graph of order $m\geq2$, $v$ is a specified vertex of $H_v$. Also let $G[F,E_k,H_{uv}]$ be the graph with $k$ edge-pockets, where $F$ is a simple graph of order $n\geq2$, $E_k=\{e_1,\ldots,e_k\}$ is a subset of the edge set of $F$ and $H_{uv}$ is a simple graph of order $m\geq3$, $uv$ is a specified edge of $H_{uv}$ such that $H_{uv}-u$ is isomorphic to $H_{uv}-v$. In this paper, we obtain some results describing the signless Laplacian spectra of $G[F,V_k,H_v]$ and $G[F,E_k,H_{uv}]$ in terms of the signless Laplacian spectra of $F,H_v$ and $F,H_{uv}$, respectively. In addition, we also give some results describing the adjacency spectrum of $G[F,V_k,H_v]$ in terms of the adjacency spectra of $F,H_v$. Finally, as an application of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs.
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