25 problems

Let $G$) be a connected graph, let $E(G)$ be its edge set, let $\deg(u)$ denote the degree of a vertex $u$, let $q(G)$ denote the spectral radius of the signless Laplacian of $G$,…

Let $\mathcal{F}$ be a finite family of graphs, let $\chi(\mathcal{F})$ denote its chromatic number, and let $\operatorname{ex}(n,\mathcal{F})$ be the maximum number of edges in an…

Let $G$ be an $n$-vertex graph, and let $q(G)$ denote its signless Laplacian spectral radius. For fixed $k\geq 3$ and sufficiently large $n$, consider graphs whose signless Laplaci…

Let $G$ be a connected graph with $n$ vertices, let $c(G)=e(G)-n(G)+\omega(G)$ be its cycle-space dimension, and let $S_2(G)$ denote the sum of the two largest signless Laplacian e…

Let $G$ be a graph with $m$ edges. A graph is $K_{r+1}$-free if it contains no complete subgraph on $r+1$ vertices. Let $q(G)$ denote the largest eigenvalue of the signless Laplaci…

Let $G$ be a connected simple graph on $n$ vertices, and let $S_k(G)$ denote the sum of the first $k$ largest signless Laplacian eigenvalues of $G$. Ashraf et al.'s conjecture. For…

Let $G$ be a connected simple graph on $n$ vertices, let $S_2(G)$ denote the sum of its two largest signless Laplacian eigenvalues, and define … Let $K^+_{1,n-1}$ denote the star g…

Signless Brouwer conjecture. For an integer $k$ with $1\leq k\leq n$,

Chen–Zhang's conjecture.

Cvetković–Rowlinson–Simić conjecture. Over all connected graphs on $n\geq6$ vertices, $s_Q(G)$ is maximized by $K_{n-1}^+$ and minimized by $P_n$ and, when $n$ is odd, by $C_n$.

For integers $k\ge 1$ and $r\ge 3$, let $F_{k,r}$ be the forbidden graph from the source, let $S_{n,a}=K_a\vee I_{n-a}$, and let $q(G)$ denote the signless Laplacian spectral radiu…

Let $G$ be a graph containing no copy of $K_{r+1}$, and let $m$ be its number of edges. Write $q(G)$ for the signless Laplacian spectral radius. Bollobás–Nikiforov conjecture. One…

Let $G$ be a graph with $n\geq5$ vertices and $3\leq k\leq n-2$, let $Q(G)$ be its signless Laplacian matrix, and let $S_k(Q(G))$ denote the sum of its $k$ largest eigenvalues. Def…

Let $G$ be a graph with $n$ vertices and $e(G)$ edges, let $Q(G)$ be its signless Laplacian matrix, and let $S_k(Q(G))$ denote the sum of the $k$ largest eigenvalues of $Q(G)$. Ash…

Signless Laplacian extremal conjecture.

Let $G$ be a graph on $n$ vertices, and let $\lambda_2(G)$ denote its second signless Laplacian eigenvalue. Brondani–de Lima–Oliveira conjecture. … This conjecture slightly improve…

Let $S_{2}(m,d,k)$ be the $k$-uniform supertree obtained from a loose path of diameter $d$ by attaching $m-d$ edges at the vertex $v_{\lfloor d/2\rfloor}$. Let $\mathbb{S}(m,d,k)$…

Let $G$ be a graph with order $n$ and size $m$. de Lima–Oliveira–de Abreu–Nikiforov conjecture. … The source presents this as a conjectured lower bound for the least signless Lapla…

Let $K_w$ be the complete graph on $w$ vertices, let $C_n$ be the cycle graph on $n$ vertices, and let $mC_n$ denote the disjoint union of $m$ copies of $C_n$. Write…

You and Yang's conjecture. (i) If $0<\alpha<1$, then

You and Yang's conjecture. If $\alpha>1$, then

Lima et al.'s conjecture.

Let $\mathcal{F}_n$ be the family of graphs on $n$ vertices, let $\mathcal{V}_n^k$ denote the graphs in $\mathcal{F}_n$ with vertex connectivity at most $k$, and let $n,k$ be posit…

Bipartite signless Laplacian power-sum conjecture. If $\alpha>1$, then

Hansen–Lucas conjectures. The following inequalities and equality characterizations hold: