12 problems

Let $n$ denote the order of a connected graph, and let the spectral sum be $\lambda_1(G)+\lambda_2(G)$, where $\lambda_1(G)$ and $\lambda_2(G)$ are the largest and second-largest a…

Extremal alpha-index and algebraic-connectivity conjecture. Given fixed $n\geq k+1$ and $k\geq 2$, the unique $k$-path graph that maximizes the $\alpha$-index for $\alpha\in[0,1]$…

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 $\mathcal{G}(n,\Delta)$ be the set of connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attain the maximum spectral radius. Suppose $\Delta\geq 3$ and…

Positive $p$-energy path-minimization conjecture.

Extremal blowup conjecture. For any $j\geq 1$, we have

Let $G$ be a finite graph, let $K_q$ be the complete graph on $q$ vertices, and let $\hom(G,K_q)$ denote the number of graph homomorphisms from $G$ to $K_q$. Perkins–Perarnau's col…

Minimum spectral radius conjecture. For every graph $G\in\mathbb{G}_{n,\lfloor\frac{n}{2}\rfloor}$,

Dumbbell-like graph conjecture. Dumbbell-like graphs $D_{a,b,a'}^{\ell}$ attain the minimum value of sum-Balaban index.

Balanced dumbbell conjecture. Among all dumbbell graphs $D_{a,b,a'}$ on at least $14$ vertices, the minimum value of sum-Balaban index is achieved for one with $a'=a$ or $a'=a+1$.

The extremal $K_4$-free graph conjecture. For $n\ge 10$, the $n$-vertex graph not containing $K_4$ that maximizes $\pi_f$ is $K_{1,\lfloor (n-1)/2\rfloor,\lceil (n-1)/2\rceil}$. Th…

Let $G$ be a graph with exactly $n$ vertices and $m=2(n-2)$ edges. The complete-bipartite extremal conjecture. Among all such graphs, a graph maximizing the algebraic connectivity…