Related papers: Mostar index and bounded maximum degree

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. We say vertices $u,v\in V$ are sum distinguished if $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. By…

A graph $G$ consists of two parts, the vertices and edges. The vertices constitute the vertex set $V(G)$ and the edges, the edge set. An edge \( e=xy \), \( ev \)-dominates not only the vertices incident to it but also those adjacent to…

Let $G$ be a connected graph of order $n$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the mean of all eccentricities in $G$. We give upper bounds on the…

We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…

O and Shi proved that the Randi\'c index of any graph $G$ with minimum degree at least $\delta$ and maximum degree at most $\Delta$ is at least $\frac{\sqrt{\delta\Delta}}{\delta+\Delta}|G|$, with equality if and only if the graph is…

Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined…

The edge Szeged index of a graph $G$ is defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the…

In this paper we show that for any graph $H$ of order $m$ and any graph $G$ of order $n$ and maximum degree $\Delta$ one can compute the number of subsets $S$ of $V(G)$ that induces a graph isomorphic to $H $in time $O(c^m\cdot n)$ for some…

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity…

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the…

For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}.$ We show that…

The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on $n$ vertices with fixed number of blocks $p$.…

Recently, a couple of degree-based topological indices, defined using a geometrical point of view of a graph edge, have attracted significant attention and being extensively investigated. Furtula and Oz [Complementary Topological Indices,…

The Wiener index of a strong digraph $D$ is defined as the sum of the distances between all ordered pairs of vertices. This definition has been extended to digraphs that are not necessarily strong by defining the distance from a vertex $a$…

The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic…

For a molecular graph, the first multiplicative Zagreb index $\Pi_1$ is equal to the product of the square of the degree of the vertices, while the second multiplicative Zagreb index $\Pi_2$ is equal to the product of the endvertex degree…

We study the component structure of the random graph $G=G_{n,m,d}$. Here $d=O(1)$ and $G$ is sampled uniformly from ${\mathcal G}_{n,m,d}$, the set of graphs with vertex set $[n]$, $m$ edges and maximum degree at most $d$. If $m=\mu n/2$…

In this paper, we investigate the Diminished Sombor index (DSO), a recently introduced degree-based topological index for a simple graph $G$, defined as \[ DSO(G) = \sum_{uv \in E} \frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}, \] where $d_u$ denotes…

Given a graph $G$, let $\mu(G)$ denote the size of the smallest maximal independent set in $G$. A family of subsets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot…

A subset $S$ of vertices in a graph $G$ is a secure total dominating set of $G$ if $S$ is a total dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup…