arXiv subjects

Đorđe Mitrović

Publications and source records attributed to Đorđe Mitrović.

6 recordsLinked to original sources

arXiv2026

Asymptotical results on the automorphism groups of canonical double cover of graphs

The canonical double cover $BX$ of a graph $X$ is the direct product $X\times K_2$. A graph $X$ is stable if every automorphism of $BX$ is induced by automorphisms of $X$ and $K_2$, and unstable otherwise. We prove several asymptotic results on stable and unstable graphs. In particular, we show that the automorphism group of the canonical double cover of almost every graph has order $2$, and consequently that almost all graphs are stable. We also prove that the dominant sources of instability are having twins and being disconnected, in that order, and that graphs arising from a known construction are the next step in this hierarchy.

math.CO↗

arXiv2025

Closing paths to cycles in symmetric graphs

It was shown by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in edge-transitive graphs, where the cycle can be assumed to be induced if the path is induced. Motivated by these results, we consider the following problem: For a given class of graphs, determine all integers $\ell\geq 0$ such that for every graph in the class, every path of length at most $\ell$ closes to a cycle. We also consider the variant of the problem for induced paths closing to induced cycles. We completely solve these problems for the classes of (finite) vertex-transitive graphs, edge-transitive graphs, and edge-transitive graphs that are not stars. For all but one case of a negative answer, we provide infinite families of connected counterexamples.

math.CO↗

arXiv2025

On transitive permutation groups with exponential graph growth

Let $Γ$ be a finite connected graph and $G$ a vertex-transitive group of its automorphisms. The pair $(Γ, G)$ is said to be locally-$L$ if the permutation group induced by the action of the vertex-stabiliser $G_v$ on the set of neighbours of a vertex $v$ in $Γ$ is permutation isomorphic to $L$. The maximum growth of $|G_v|$ as a function of $|VΓ|$ for locally-$L$ pairs $(Γ,G)$ is called the graph growth of $L$. We prove that if $L$ is a transitive permutation group on a set $Ω$ admitting a nontrivial block $B$ such that the pointwise stabiliser of $Ω\setminus B$ in $L$ is nontrivial, then the graph growth of $L$ is exponential. This generalises several results in the literature on transitive permutation groups with exponential graph growth.

math.CO↗

arXiv2022

Some conditions implying stability of graphs

A graph $X$ is said to be unstable if the direct product $X\times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is non-trivially unstable if it is unstable, connected, non-bipartite, and distinct vertices have distinct sets of neighbours. In this paper, we prove two sufficient conditions for stability of graphs in which every edge lies on a triangle, revising an incorrect claim of Surowski and filling in some gaps in the proof of another one. We also consider triangle-free graphs, and prove that there are no non-trivially unstable triangle-free graphs of diameter 2. An interesting construction of non-trivially unstable graphs is given and several open problems are posed.

math.CO↗

arXiv2021

On automorphisms of the double cover of a circulant graph

A graph $X$ is said to be "unstable" if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially unstable" if it is unstable, connected, and nonbipartite, and no two distinct vertices of X have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

math.CO↗

arXiv2021

Automorphisms of the double cover of a circulant graph of valency at most 7

A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most $7$. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, because there is a nontrivially unstable circulant graph of valency $8$ that does not satisfy the hypotheses of any of Wilson's four instability theorems for circulant graphs.

math.CO↗