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The study of Markov processes and broadcasting on trees has deep connections to a variety of areas including statistical physics, graphical models, phylogenetic reconstruction, Markov Chain Monte Carlo, and community detection in random…

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The…

For a simple graph $G(V,E)$, the zeroth-order general Randi\' c index is defined as $^0R_{\alpha}(G)=\sum_{v\in V(G)}d(v)^{\alpha}$, where $d(v)$ is the degree of the vertex $v$ and $\alpha\ne0$ is a real number. The $k$-generalized…

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…

The Barat-Thomassen conjecture, recently proved in [Bensmail et al.: A proof of the Barat-Thomassen conjecture. J. Combin. Theory Ser. B, 124:39-55, 2017.], asserts that for every tree T, there is a constant $c_T$ such that every $c_T$-edge…

In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity…

Hasunuma [J. Graph Theory 102 (2023) 423-435] conjectured that for any tree $T$ of order $m$, every $k$-connected (or $k$-edge-connected) graph $G$ with minimum degree at least $k+m-1$ contains a tree $T'\cong T$ such that $G-E(T')$ is…

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…

Let $G=(V,E)$ be a simple, unweighted, connected graph. Let $d(u,v)$ denote the distance between vertices $u,v$. A resolving set of $G$ is a subset $S$ of $V$ such that knowing the distance from a vertex $v$ to every vertex in $S$ uniquely…

The component size of a graph is the maximum number of edges in any connected component of the graph. Given a graph $G$ and two integers $k$ and $c$, $(k,c)$-Decomposition is the problem of deciding whether $G$ admits an edge partition into…

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node connectivity. They have applications in computer science, network science,…

In this paper we describe an algorithm that embeds a graph metric $(V,d_G)$ on an undirected weighted graph $G=(V,E)$ into a distribution of tree metrics $(T,D_T)$ such that for every pair $u,v\in V$, $d_G(u,v)\leq d_T(u,v)$ and…

In this paper we present various lower bound results on collective tree spanners and on spanners of bounded treewidth. A graph $G$ is said to admit a system of $\mu$ collective additive tree $c$-spanners if there is a system $\cal{T}$$(G)$…

The status of a vertex $v$ in a connected graph is the sum of the distances from $v$ to all other vertices. The status sequence of a connected graph is the list of the statuses of all the vertices of the graph. In this paper we investigate…

The edge-Szeged index of a graph $G$ is defined as $Sz_e(G) = \sum_{e=uv \in E(G)}m_u(e)m_v(e)$, where $m_u(e)$ denotes the number of edges of $G$ whose distance to $u$ is smaller than the distance to $v$ and $m_v(e)$ denotes the number of…

The sigma index in graph theory refers to a measure of the degree differences between vertices in a graph. The goal is to determine the graphs that have the maximum sigma index within certain classes of graphs. Abdo, Dimitrov, and Gutman…

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a…

Let $G=(V,E)$ be a simple undirected and connected graph on $n$ vertices. The Graovac--Ghorbani index of a graph $G$ is defined as $$ABC_{GG}(G)= \sum_{uv \in E(G)} \sqrt{\frac{n_{u}+n_{v}-2} {n_{u} n_{v}}},$$ where $n_u$ is the number of…

A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no…

Previous studies has shown that for a weighted undirected graph having $n$ vertices and $m$ edges, a minimal weight spanning tree can be found with $O^*(\sqrt{mn})$ calls to the weight oracle. The present note shows that a given spanning…