Related papers: On the conjecture about the exponential reduced So…

The zeroth-order general Randi\'c index (usually denoted by $R_{\alpha}^{0}$) and variable sum exdeg index (denoted by $SEI_{a}$) of a graph $G$ are defined as $R_{\alpha}^{0}(G)= \sum_{v\in V(G)} (d_{v})^{\alpha}$ and $SEI_{a}(G)=…

The domination number $\gamma(G)$ of a graph $G$, its exponential domination number $\gamma_e(G)$, and its porous exponential domination number $\gamma_e^*(G)$ satisfy $\gamma_e^*(G)\leq \gamma_e(G)\leq \gamma(G)$. We contribute results…

The weakly zero-divisor graph $W\Gamma(R)$ of a commutative ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two distinct vertices $x$, $y$ are adjacent if and only if there exist $w\in {\rm…

From any directed graph $E$ one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in $E$. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for…

The sigma-irregularity index $\sigma(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $\sigma(T)$ within the class of trees of fixed order $n$ and…

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two…

Let $G$ be a connected graph with $n$ vertices and $m$ edges. The vertex-degree-based topological index (VDB) (or graphical function-index) $TI(G)$ of $G$ with edge-weight function $I(x,y)$ is defined as $$TI(G)=\sum\limits_{uv\in…

The second Zagreb index is $M_2(G)=\sum_{uv\in E(G)}d_{G}(u)d_{G}(v)$. It was found to occur in certain approximate expressions of the total $\pi$-electron energy of alternant hydrocarbons and used by various researchers in their QSPR and…

A large number of graph invariants of the form $\sum_{uv \in E(G)} F(d_u,d_v)$ are studied in mathematical chemistry, where $uv$ denotes the edge of the graph $G$ connecting the vertices $u$ and $v$, and $d_u$ is the degree of the vertex…

The resolvent energy of a graph $G$ of order $n$ is defined as $ER=\sum_{i=1}^n (n-\lambda_i)^{-1}$, where $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $G$. In a recent work [Gutman et al., {\it MATCH Commun. Math. Comput.…

A vertex-degree-based topological index named as Sombor index of a simple graph G with n vertices was recently introduced by I. Gutman. In this paper, we find Sombor index of m-splitting graph and m-shadow graph. Also, we determine relation…

Given a graph $G$ and $p \in [0,1]$, let $G_p$ denote the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Alon, Krivelevich, and Sudokov proved $\mathbb{E} [\chi(G_p)] \geq C_p \frac{\chi(G)}{\log…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type…

For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by $\varepsilon(G)$, is the sum of the eccentricity of individual vertices. In…

Gomory and Hu proved that if $ G $ is a finite graph with non-negative weights on its edges, then there exists a tree $ T $ (called now Gomory-Hu tree) on $ V(G) $ such that for all $ u\neq v\in V(G) $ there is an $ e\in E(T) $ such that…

A well-known result due to Caro (1979) and Wei (1981) states that every graph $G$ has an independent set of size at least $\sum_{v\in V(G)} \frac{1}{d(v) + 1}$, where $d(v)$ denotes the degree of vertex $v$. Alon, Kahn, and Seymour (1987)…

For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $\lambda_1(G) + \lambda_2(G)$, where $\lambda_1(G)$ (resp. $\lambda_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar,…

For any graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges, and a positive integer $r$, by $\chi'_{\Sigma,r}(G)$ we denote the $r$-distant sum distinguishing index of $G$. This is the least integer $k$ for which a…

Let $G$ be a connected graph. The revised edge Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is…