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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…

Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for…

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…

As a variant of the much studied Tur\'an number, $ex(n,F)$, the largest number of edges that an $n$-vertex $F$-free graph may contain, we introduce the connected Tur\'an number $ex_c(n,F)$, the largest number of edges that an $n$-vertex…

The Randic (connectivity) index is one of the most successful molecular descriptors in structure-property and structure-activity relationships studies. J. Gao found the sharp upper bound for the Randic index of apex trees. In this paper, we…

A set of novel vertex-degree-based invariants was introduced by Gutman, denoted by \newline $SO_1, SO_2, \ldots,SO_6$. These invariants were constructed through geometric reasoning based on a new graph invariant framework. Motivated by…

In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been…

In 1989, Zehavi and Itai conjectured that every $k$-connected graph contains $k$ independent spanning trees rooted at any prescribed vertex $r$. That is, for each vertex $v$, the unique $r$-$v$ paths within these $k$ spanning trees are…

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

Vertex-degree-based topological indices have recently gained a lot of attention from mathematical chemists. One such index that we focus on in this paper is called Sombor index. After its definition in late 2020, the Sombor index was…

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if…

Diestel and M\"uller showed that the connected tree-width of a graph $G$, i.e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of $G$ and the largest length of a geodesic cycle in…

We investigate the tractability of a simple fusion of two fundamental structures on graphs, a spanning tree and a perfect matching. Specifically, we consider the following problem: given an edge-weighted graph, find a minimum-weight…

A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if no two edges of $T$ are assigned the same color. For a vertex subset $S\in V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. A {\em $k$-rainbow…

Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the…

For a simple graph $G=(V,E),$ let $\mathcal{S}_+(G)$ denote the set of real positive semidefinite matrices $A=(a_{ij})$ such that $a_{ij}\neq 0$ if $\{i,j\}\in E$ and $a_{ij}=0$ if $\{i,j\}\notin E$. The maximum positive semidefinite…

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…

The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report…

Bae and Park found an upper bound on the arc index of prime links in terms of the minimal crossing number. In this paper, we extend the definition of the arc presentation to spatial graphs and find an upper bound on the arc index $\alpha…