17 problems

Chemical antiregular graph conjecture. The same graphs as in Theorem $$ have maximum value of $\sigma_t^{f(n)}$ even if $f(n)$ is a constant in the interval $(0,1)$.

Strong conjecture on rooted $F$-irregular graphs. For every connected graph $F$ of order $|F| \ge 3$ and each root $r$ of $F$, unless $F=K_{1,n-1}$ and $r$ is the center of…

Chartrand–Holbert–Oellermann–Swart conjecture. For every connected graph $F$ of order at least $3$, there exists a nontrivial $F$-irregular graph.

For a digraph $D$, define the balanced degree of a vertex $x$ by … A digraph is strongly locally irregular if $\sigma(x)\neq\sigma(y)$ for every arc $xy$, and let…

For a digraph $D$, two adjacent vertices are weakly distinguished when their outdegree–indegree pairs differ. Let $\mathrm{lir}(D)$ be the minimum number of colors in an arc colori…

A sink-source path is an oriented path containing an arc directed into a vertex and an arc directed out of that vertex, as used in the source. For $a,b\in\{+,-\}$, define $(a,b)$-l…

For $a,b\in\{+,-\}$, a digraph is $(a,b)$-locally irregular if, for every arc $uv$, $d^a(u)\neq d^b(v)$; let $\mathrm{lir}^{(+,+)}(D)$ be the minimum number of colors in an arc col…

Let $G$ be a connected locally irregular colorable graph, let $B$ be the bow-tie graph, and let $\mathrm{lir}(G)$ denote the minimum number of locally irregular subgraphs whose edg…

Tree irregularity inequality conjecture.

Triangle-free extremal conjecture. Among connected triangle-free graphs on $n$ vertices, the extremal complete bipartite graphs have the maximum $\sigma_t(G)$-irregularity.

Barja–González–Naranjo's modified conjecture. For every non-isotrivial fibred surface $f\colon S\to B$ of genus $g\geq 2$,

Invariant-property conjecture. For any pair $(\gamma,\rho)$, there exists a property $\mathcal{P}$ such that:

Let $f\colon S\to B$ be a fibred surface, meaning a fibration from a smooth projective surface $S$ to a smooth projective curve $B$, with general fibre of genus $g\geq 2$. Write…

Elphick's conjecture. The graph energy satisfies

Let $G$ be a simple connected graph with $n$ vertices, and let $\operatorname{irr}_{t}(G)$ denote its total irregularity. A graph is regular if all its vertices have the same degre…

Modified Xiao's conjecture. Every non-trivial fibration satisfies

Let $f:S\rightarrow B$ be a non-trivial fibration from a smooth projective surface $S$ to a smooth projective curve $B$, with fibres of genus $g$. Write $q_f=q(S)-g(B)$ for its rel…