With geometric significance, the Euler Sombor index of a graph Γ is defined as EP(Γ)=∑{uv}∈E(Γ)d(u)2+d(v)2+d(u)d(v). It originates from mathematical distance property and has been proven to have good chemical applications in octane isomers. In this paper, minimum maximum for unicyclic graphs with given girth, well corresponding extremal graphs, are determined. As an application, experimental values some benzenoid hydrocarbons drug molecules were compared boiling point. Through regression...

Topological indices are widely used molecular structure descriptors in chemistry and pharmaceutics, which help analyze predict the physicochemical properties biological activity of compounds. Focusing on discriminative power, researchers introduced general exponential VDB topological index, is defined as follows: . Our current work focuses studying some using methods. In paper, we characterize sufficient conditions for that (1) graph M n , k minimal among unicyclic with girth (2) H maximal...

In this paper, we first show that a result of Girela et al. on analytic functions can be extended to hyperbolic-harmonic functions, and then establish Hardy-Littlewood-type theorems hyperbolic harmonic functions.

In this paper, a descent Liu–Storey conjugate gradient method is extended to solve large-scale nonlinear systems of equations. Based on certain assumptions, the global convergence property obtained with nonmonotone line search. The proposed suitable problems for low-storage requirement. Numerical experiment results show that new practically effective.

<abstract><p>For a graph $ G $, let n_G(u, v) be the number of vertices that are strictly closer to u than v $. The distance–unbalancedness index {\rm uB}(G) is defined as sum |n_G(u, v)-n_G(v, u)| over all unordered pairs and In this paper, we show minimum merged C_3\cdot T (n+2)(n-3) where C_3 \cdot obtained by attaching tree cycle $.</p></abstract>

Vertex-degree-based (VDB) topological indices have been applied in QSPR/QSAR. As an important category, the general logarithmic VDB index $T_{lnf}(G)$, is defined as summation of $ln{f(d(u),d(v))}$, where over all $uv\in E(G)$. In this paper, we give sufficient conditions for that (1) path $P_{n}$ only tree with minimal $T_{lnf}$; (2) star $S_n$ maximal and $T_{lnf}$, respectively. applications, trees some are determined.

Vertex-degree-based (VDB) topological indices have been applied in the study of molecular structures and chemical properties. At present, exponential VDB index has studied extensively. Naturally, we began to consider logarithmic lnTf. In this paper, first discuss necessity a index, then present sufficient conditions so that Pn Sn are only trees with smallest greatest values lnTf(T). As applications, minimal maximal some determined. Through our work, found lnTf excellent discriminability, but...

The crossing numbers of graphs were started from Turán’s brick factory problem (TBFP). Because its wide range applications, it has been used in computer networks, electrical circuits, and biological engineering. Recently, many experts began to pay much attention the number <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>G</mi> <mo>\</mo> <mi>e</mi> </math> , which obtained graph id="M2"> by deleting an edge id="M3"> . In this paper, using some combinatorial skills, we determine...

The Wiener index of a connected graph is the sum distances between all pairs vertices.Let G(n, 2k) be set graphs on n vertices with exactly 2k odd degree.G(n, 0) just Eulerian vertices.In Gutman et al. (2014) and Dankelmann (2021), authors characterized first four minimum two maximum index.In paper, we characterize in for 0 ≤ k 2 , 2) first-maximum secondmaximum index.

&lt;abstract&gt;&lt;p&gt;For an integer $ h\geq 0 $, the h $th order Randić index for a simple graph G is defined as R^{h}(G) = \sum_{\pi}\frac{1}{\sqrt{v_1(\pi)v_2(\pi)\cdots v_{h+1}(\pi)}} where \pi extends over all paths of length in and v_i(\pi) denotes degree i $-th vertex path $. In this paper, we showed that R^{h}(T) double starlike tree T (a with two vertices degrees m_1, m_2 &amp;gt; 2 $) completely determined by its branches \leq As consequence, proved trees equal $-Randić are...

The Harary index is one of the most investigated topological indices in mathematical chemistry and related disciplines, which can be used to describe molecular structure physical properties. Let <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi mathvariant="double-struck">G</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>n</mi> <mo>,</mo> <mn>2</mn> <mi>k</mi> </mrow> </mfenced> </math> set all connected graphs on n vertices with exactly id="M2"> odd degree. Recently,...