Related papers: Bounds on connective constants of regular graphs

Let G be a finite graph with the non-k-order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). A major result of the paper applies model-theoretic arguments to obtain a stronger version of…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of $I(G)^q$ in terms of certain combinatorial invariants associated with $G$. We…

Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$,…

The longstanding open problem of approximating all singular vertex couplings in a quantum graph is solved. We present a construction in which the edges are decoupled; an each pair of their endpoints is joined by an edge carrying a $\delta$…

An edge subset \( S \subseteq E(G) \) is called a 3-restricted edge-cut if \( G - S \) is disconnected and each component of \( G - S \) contains at least three vertices. The 3-restricted edge-connectivity of a graph \( G \), denoted by \(…

A centrality measure of the cut-edges of an undirected graph, given in [Altafini et al.~SIMAX 2023] and based on Kemeny's constant, is revisited. A numerically more stable expression is given to compute this measure, and an explicit…

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph is distance-biregular when it is…

Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph G is distance-regular if and only if its spectral excess (a number that can be…

The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) <= 2c(G) - 1…

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ and $D\geq 3$. Assume the intersection numbers $a_1=0$ and $a_2\not=0$. We show $\Gamma$ is 3-bounded in the sense of the article [D-bounded…

A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all…

A graph $G = (V, E)$ of bounded degree has an adjacency operator~$A$ which acts on the Hilbert space $\ell^2(V)$. There are different kinds of measures of interest on the spectrum $\Sigma (A)$ of $A$. In particular, each vector $\xi \in…

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…

A set $Z$ of vertices of a graph $G$ is a zero forcing set of $G$ if initially labeling all vertices in $Z$ with $1$ and all remaining vertices of $G$ with $0$, and then, iteratively and as long as possible, changing the label of some…

A $k$-$\gamma_{c}$-edge critical graph is a graph $G$ with the connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be…

In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to…

Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G…

We say a directed graph $G$ on $n$ vertices is irredundant if the removal of any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such that there exists a directed path from $u$ to $v$. We determine the maximum possible…