Related papers: All Terminal Reliability Roots of Smallest Modulus

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce…

A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is $2$-connected and cubic, then any longest cycle must have a chord. He also showed that if $G$…

The matching polynomial of a graph encodes rich combinatorial information through its roots. We determine the maximum multiplicity of a non-zero matching polynomial root and characterize all graphs attaining the bound. We also generalize…

Let $G$ be a connected graph on $n$ vertices. The Gallai number $Gal(G)$ of $G$ is the size of the smallest set of vertices that meets every maximum path in $G$. Gr\"unbaum constructed a graph $G$ with $Gal(G)=3$. Very recently, Long,…

We consider growing random networks $\{\mathcal G_n\}_{n \ge 1}$ where, at each time, a new vertex attaches itself to a collection of existing vertices via a fixed number $m \ge 1$ of edges, with probability proportional to an attachment…

We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both…

In this paper we deal with two aspects of the minimum rank of a simple undirected graph $G$ on $n$ vertices over a finite field $\FF_q$ with $q$ elements, which is denoted by $\mr(\FF_q,G)$. In the first part of this paper we show that the…

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special…

The ordinary generating function of the number of complete subgraphs of $G$ is called a clique polynomial of $G$ and is denoted by $C(G,x)$. A real root of $C(G,x)$ is called a clique root of the graph $G$. Hajiabolhasan and Mehrabadi…

While the problem of determining whether an embedding of a graph $G$ in $\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying whether a given embedding of $G$ is {\it rigid} or not is still a hard task that usually…

The paper studies wsat$(G,H)$ which is the minimum number of edges in a weakly $H$-saturated subgraph of $G$. We prove that wsat$(K_n,H)$ is `stable' - remains the same after independent removal of every edge of $K_n$ with constant…

Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one;…

A graph $G$ on $n$ vertices is \textit{pancyclic} if it contains cycles of length $t$ for all $3 \leq t \leq n$. In this paper we prove that for any fixed $\epsilon>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ asymptotically almost…

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$…

In the abelian sandpile model, recurrent chip configurations are of interest as they are a natural choice of coset representatives under the quotient of the reduced Laplacian. We investigate graphs whose recurrent identities with respect to…

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that…

We study a class of rooted trees with a substitution type structure. These trees are not necessarily regular, but exhibit a lot of symmetries. We consider nearest neighbor operators which reflect the symmetries of the trees. The spectrum of…

The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number of…

Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with spanning tree modulus is known to produce two hierarchical…

We define $PC$-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root $\beta(G)$ of $PC$-polynomial of the corresponding graph.…