The Sombor index is a novel vertex-degree-based topological index which was proposed by Gutman in 2021. This index has been shown to be helpful in predicting the enthalpy of vaporization and entropy ...of octane isomers. In 2022, Gutman put forward a new variant of the Sombor index and gave its geometric interpretation. This invariant which we call the geometric Sombor index can be considered as a practical tool for measuring irregularity in graphs. Our aim is to study some basic mathematical properties of this new Sombor-type invariant. Especially, we prove that for any tree (resp. unicyclic graph) with a fixed order and maximum degree
Δ
, the geometric Sombor index is bounded below by
Δ
3
-
Δ
2
(resp.
Δ
3
-
Δ
-
6
2
). Also the extremal trees and unicyclic graphs that achieve the lower bound are characterized.
For a (molecular) graph G, the first and the second entire Zagreb indices are defined by the formulas M1εG=∑x∈VG∪EGdx2 and M2εG=∑x is either adjacent or incident to ydxdy in which dx represents the ...degree of a vertex or an edge x. In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds.
Let $G$ be a finite and simple graph with vertex set $V(G)$. A nonnegative signed total Roman dominating function (NNSTRDF) on a graph $G$ is a function $f:V(G)\rightarrow\{-1, 1, 2\}$ satisfying the ...conditions that (i) $\sum_{x\in N(v)}f(x)\ge 0$ for each $v\in V(G)$, where $N(v)$ is the open neighborhood of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has a neighbor $v$ for which $f(v)=2$. The weight of an NNSTRDF $f$ is $\omega(f)=\sum_{v\in V (G)}f(v)$. The nonnegative signed total Roman domination number $\gamma^{NN}_{stR}(G)$ of $G$ is the minimum weight of an NNSTRDF on $G$. In this paper we initiate the study of the nonnegative signed total Roman domination number of graphs, and we present different bounds on $\gamma^{NN}_{stR}(G)$. We determine the nonnegative signed total Roman domination number of some classes of graphs. If $n$ is the order and $m$ is the size of the graph $G$, then we show that $\gamma^{NN}_{stR}(G)\ge \frac{3}{4}(\sqrt{8n+1}+1)-n$ and $\gamma^{NN}_{stR}(G)\ge (10n-12m)/5$. In addition, if $G$ is a bipartite graph of order $n$, then we prove that $\gamma^{NN}_{stR}(G)\ge \frac{3}{2}(\sqrt{4n+1}-1)-n$.
The third leap Zagreb index is the sum of the products of vertex degrees and second degrees. In this paper, a lower bound on the third leap Zagreb index is established, and the extremal trees ...achieving this bound are characterized.
For a graph
, let
) be the domination number,
) be the independent domination number and
) be the 2-independence number. In this paper, we prove that for any tree
of order
≥ 2, 4
) − 3
) ≥ 3
), and ...we characterize all trees attaining equality. Also we prove that for every tree
of order
≥ 2,
, and we characterize all extreme trees.
Let D = (V,A) be a finite simple directed graph (shortly, digraph). A function f : V → {−1, 0, 1} is called a twin minus total dominating function (TMTDF) if f(N−(v)) ≥ 1 and f(N+(v)) ≥ 1 for each ...vertex v ∈ V. The twin minus total domination number of D is y*mt(D) = min{w(f) | f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for y*mt(D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.
The Mostar index is a recently‐introduced molecular structure descriptor used for measuring the peripherality and distance‐non‐balancedness extent in molecular graphs. In this paper, we present exact ...formulae or sharp upper bounds on the Mostar index of some families of graph operations. Using the obtained results, the peripherality extent of some chemical graphs and nanostructures are measured and a number of results related to the Mostar index of graph operations reported in (Int. J. Quantum. Chem. 2021, 121(15), e26674) and (TWMS J. App. and Eng. Math. 2021, 11(2), 587) are improved or generalized.
Studying topological indices of molecular graphs related to chemical compounds is difficult and time consuming work in general. One of the efficient ways for such studies is to produce molecular graphs with complicated structures from simpler graphs by using operations on graphs. This paper is concerned with studying a recently‐introduced molecular structure descriptor called Mostar index for various families of graph operations and some chemical graphs and nanostructures constructed from them.
We investigate some relationships between two vastly studied parameters of a simple graph $G$. These parameters include mixed domination number (denoted by $\gamma_m(G)$) and 2-independence number ...($\beta_2(G)$). For a tree $T$, we obtain $\frac{3}{4}\beta_2(T)\ge \gamma_m(T)$ and characterized all those trees which attain the equality.
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