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An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized Bethe tree with $k$ levels is a rooted tree in which vertices at the same level have…

In this note, we propose a conjecture stating that some series involving primitive sequences are convergent. Then, we show (by a counterexample) that the analogue of a conjecture of Erd\H{o}s, for those series, is false.

For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree…

Given a bridgeless graph G, the well-known cycle double cover conjecture posits that there is a list of cycles of G, such that every edge appears in exactly two cycles. In this paper, we prove the cycle double cover conjecture. More…

Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be the number of connected spanning subgraphs of a connected graph $G$. We bound $F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree. Among…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…

We prove the Baum--Connes conjecture with arbitrary coefficients for some classes of groups: (1) Linear algebraic groups over a non-archimedean local field. (2) Linear algebraic groups over the adeles of a global field k, provided that at…

Recently, Milani\v{c} and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight $1$ if and only if it forms a maximal…

We compare two notions of $G$-fiber bundles and $G$-principal bundles in the literature, with an aim to clarify early results in equivariant bundle theory that are needed in current work of equivariant algebraic topology. We also give…

A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order…

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…

We prove the Strong Nine Dragon Tree Conjecture is true if we replace the edge bound with $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil \leq d +…

Let ${\rm Z}(G)$ and ${\rm gp}(G)$ be the zero forcing number and the general position number of a graph $G$, respectively. Known results imply that ${\rm gp}(T)\ge {\rm Z}(T) + 1$ holds for every nontrivial tree $T$. It is proved that the…

Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of…

Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges,…

We propose a new conjecture on some exponential sums. These particular sums have not apparently been considered in the literature. Subject to the conjecture we obtain the first effective construction of asymptotically good tree codes. The…

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$…

If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights…

It is easy to imagine that a subvariety of a vector bundle, whose intersection with every fibre is a vector subspace of constant dimension, must necessarily be a sub-bundle. We give two examples to show that this is not true, and several…

This paper proposes a generalized ABC conjecture and assuming its validity settles a generalized version of Fermats last theorem.