11 problems

Let $\alpha_{1,k}$ denote the asymptotic extremal constant studied in the paper for the relevant sums of adjacency eigenvalues of a graph and its complement. Proposition gives a lo…

Let $G$ be a graph on $n$ vertices, and let $\overline G$ denote its complement. Write $\lambda_1(G)$ for the largest adjacency eigenvalue of $G$. Nikiforov's conjecture. One has ……

For an integer $k\geq 3$, let $\alpha_{1,k}$ be the asymptotic extremal constant studied in the paper, and let $J_r$ and $K_r$ denote the indicated graph constructions. Extremal co…

Let $G$ be a graph on $n$ vertices. A dominating set is a subset $S$ of the vertices such that every vertex is either in $S$ or adjacent to a vertex in $S$, and let $\bar{G}$ denot…

Stevanović's conjecture. The maximum value of

Let $G$ be a graph with $n$ vertices, let $\overline{G}$ be its complement, and let $s_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of $G$. Nordhaus–Gaddum full Br…

Let $G$ be a graph of order $n\geq 2$, and let $\mu_1(G)\geq\cdots\geq\mu_{n-1}(G)$ be its Laplacian eigenvalues. The Laplacian spread is $\mu_1(G)-\mu_{n-1}(G)$. Zhai–Shu–Hong con…

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…

Harary–Haynes conjecture.

Let $G$ be a simple graph on $n\geq 2$ vertices, let $\overline G$ be its complement, and let $q_1(H)$ denote the largest signless Laplacian eigenvalue of a graph $H$. Aouchiche–Ha…

Let $G$ be a graph with $n$ vertices, let $\overline G$ be its complement, and let $K_1$ denote the one-vertex complete graph. The join of graphs is denoted by $\vee$. Laplacian pr…