Related papers: On Zero-free Intervals of Flow Polynomials

A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic…

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem…

The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z \in \mathbb{C} : \mathrm{Re}(z) \leq 0\}$, and the graph itself is also referred to as stable. Brown and Cameron…

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is called \textit{$\mathcal{F}$-free} if for any $F\in \mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized…

We prove that the \emph{standard zero forcing number} $Z(G)$ and the \emph{positive semidefinite zero forcing number} $Z_+(G)$ are equal for all claw-free graphs $G$. This result resolves a conjecture proposed by the computer program…

A graph $G$ is a $k$-leaf power if there exists a tree $T$ whose leaf set is $V(G)$, and such that $uv \in E(G)$ if and only if the distance between $u$ and $v$ in $T$ is at most $k$. The graph classes of $k$-leaf powers have several…

Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $\phi$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $\lambda(\phi)$, which is a…

Let $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an…

A $k$-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than $k$ in the graph. The $k$-independence number of a graph, denoted $\alpha_k$, is the size of a largest…

Bollob\'as and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $\lambda^2_1(G)+\lambda^2_2(G)\leq \frac{r-1}{r}\cdot2m$,…

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory…

The $Q$-index of graph $G$ is the largest eigenvalue of the signless Laplacian matrix of $G$. Wang [Discrete Appl. Math. 356(2024)] proved the sharp upper bounds on the $Q$-index of leaf-free graphs with given size and characterized the…

Let $\{G_M\}_{M\geq 0}$ be the random graph process, where $G_0$ is the empty graph on $n$ vertices and subsequent graphs in the sequence are obtained by adding a new edge uniformly at random. For each $\varepsilon>0$, we show that, almost…

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$,…

The class of graphs that do not contain an induced path on $k$ vertices, $P_k$-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of $P_k$-free graphs, including…

For graphs $G$ and $T$, and a family of graphs $\mathcal{F}$ let $\mathrm{ex}(G,T,\mathcal{F})$ denote the maximum possible number of copies of $T$ in an $\mathcal{F}$-free subgraph of $G$. We investigate the algorithmic aspects of…

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For…

A $hole$ is an induced cycle of length at least four, and an odd hole is a hole of odd length. A {\em fork} is a graph obtained from $K_{1,3}$ by subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd hole by…

A graph is called claw-free if it contains no induced subgraph isomorphic to the complete bipartite graph $K_{1, 3}$. The undirected power graph of a group $G$ has vertices the elements of $G$, with an edge between $g_1$ and $g_2$ if one of…

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions…