Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be graph. Then, the of G is defined as SO(G)=?uv?E(G)dG2(u)+dG2(v). In this paper, we give some lemmas that can used to compare indices between two graphs. With these lemmas, determine graph with maximum SO among all cacti n vertices and k cut edges. Furthermore, unique p pendant...

Let h ≥ 1 be an integer. An L(h, 1, 1)-labelling of a (finite or infinite) graph is an assignment of nonnegative integers (labels) to its vertices such that adjacent vertices receive labels with difference at least h, and vertices distance 2 or 3 apart receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over a...

Given any graph $H$, a $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges with $q$ colors yields monochromatic subgraph isomorphic $H$. Such minimal additionally no proper $G'$ In 1976, Burr, Erdös, and Lovász initiated study parameter $s_q(H)$, defined as smallest minimum degree among all graphs this paper, we consider problem determining how many vertices $s_q(H)$ can contain...

Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ sum $k$ largest absolute values eigenvalues $G$. $C_n^+$ with $n+1$ vertices obtained from directed cycle $C_n$ by attaching pendant arc whose tail is on $C_n$. A $\mathbb{C}_n^+$-free if it contains no $C_{\ell}^+$ as subdigraph for any $2\leq \ell \leq n-1$. In this paper, we present lower bounds $S_n(G)$ digraphs order $n$. We ...

The size Ramsey number of a graph H is defined as the minimum edges in G such that there monochromatic copy every two-coloring E(G). was introduced by Erdős, Faudree, Rousseau, and Schelp 1978 they ended their foundational paper asking whether one can determine up to constant factor numbers three families graphs: complete bipartite graphs, book graphs (obtained adding many common neighbors vert...

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such can be built inductively starting from a single vertex by repeatedly adding either pendant vertices or twins (i.e., with the same neighborhood as an existing one). Dirac Duffin’s 2–connected series–parallel edge edges in series parallel. In this paper we give elementary proof two construct...

We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b \in \mathbb{R}$, $DD_{a,b} (G) = \sum_{ v V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is vertex set $G$, $deg_G (v)$ $v$, $S^b_{G} (v) w V(G) \setminus \{ \} } [d_{G} (v,w) ]^{b}$ $d_{G} (v,w)$ between $v$ $w$ in present some sharp bounds on for multipartite graphs trees g...

A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pe...