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Let G = (V (G) , E (G)) be a simple connected graph where V (G) is a vertex set and E (G) is an edge set. The Sombor index of graph G is a vertex-degree-based topological index, which is introduced by Gutman et al. and defined as S O (G) = ∑ u v ∈ E (G) d 2 (u) + d 2 (v) where d (u) denotes the degree of vertex u. In this paper, we give the extremal value of the Sombor index among all maximal outerplanar graphs with given order n and characterize those maximal outerplanar graphs with the extremal value. [ABSTRACT FROM AUTHOR]

Based on the geometric background of Sombor index and motivating by the higher order connectivity index and the Sombor index, we introduce the path-coordinate of a path in a graph and a degree-point in a higher dimensional coordinate system, and define the higher order Sombor index of a graph. We first consider mathematical properties of the higher order Sombor index, show that the higher order connectivity index of a starlike tree is completely determined by its branches and that starlike trees with a given maximum degree which have the same higher order Sombor indices are isomorphic. Then, we determine the extremal values of the second order Sombor index for all trees with n vertices and characterize the corresponding extremal trees. Finally, the chemical importance of the second order Sombor index is investigated and it is shown that the new index is useful in predicting physicochemical properties with high accuracy compared to some well-established indices. [ABSTRACT FROM AUTHOR]

I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by S O 1 , S O 2 , ... , S O 6 . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal values of the graph invariants S O 5 and S O 6 among all trees and molecular trees of order n, and characterize the trees and molecular trees that achieve the extremal values, respectively. Thus, the problem is completely solved. [ABSTRACT FROM AUTHOR]

This article is concerned with the spectral problem { d ϕ 1 (x) = ϕ 2 (x) d h (x) , d ϕ 2 (x) = ϕ 3 (x) d g (x) , d ϕ 3 (x) = − λ ϕ 1 (x) d x , x ∈ (− 1 , 1) , ϕ 2 (− 1) = 0 , ϕ 3 (− 1) = 0 , ϕ 3 (1) = 0 , which is associated with the two-component Novikov equation. The coefficients g and h are functions of bounded variation. The first aim is to prove the continuous dependence of the n -th eigenvalue on the coefficients g and h with different topologies; in addition, we discuss the Fréchet differentiability of the algebraically simple eigenvalues on the coefficients g , h with respect to the norm topology of total variations. Secondly, utilizing the theory of oscillatory (non-symmetric) kernels, conditions on g , h under which the spectrum is nonnegative and algebraically simple are obtained. Finally, we solve the extremal problems of the lowest nonzero eigenvalue, when the norm ‖ ⋅ ‖ V of the coefficients g , h is given. The latter can be seen as an application of our first two main results and the trace formulas. [ABSTRACT FROM AUTHOR]

A novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors, which is the perimeter of an ellipse whose focal points represent the degree‐point and its dual‐point of a pair of adjacent vertices in a graph G=(V,E)$$ G=\left(V,E\right) $$. We first analyze the chemical applicability of this elliptic Sombor index, and then establish its mathematical properties and determine the extremal values for this elliptic Sombor index among all (molecular) trees and characterize the corresponding extremal graphs. [ABSTRACT FROM AUTHOR]

The Gutman index and Schultz index of a connected graph are degree-distance-based topological indices. In this paper, we devoted to establish the explicit analytical expressions for the simple formulae of the expected values of the Gutman and Schultz indices in a random polygonal. Based on these results above, we get the extremal values and average values of Gunman and Schultz indices of all polygonal chains.

Let G = (V G , E G) be a simple connected graph with its vertex set V G and edge set E G . The Mostar index Mo(G) was defined as M o (G) = ∑ e = u v ∈ E (G) | n u - n v | , where n u (resp., n v ) is the number of vertices whose distance to vertex u (resp., v) is smaller than the distance to vertex v (resp., u). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion. [ABSTRACT FROM AUTHOR]

Let $ G $ be a simple connected graph with the vertex set $ V(G) $ and $ d_{B}(u, v) $ be the biharmonic distance between two vertices $ u $ and $ v $ in $ G $. The biharmonic index $ BH(G) $ of $ G $ is defined as $ BH(G) = \frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u, v) = n\sum\limits_{i = 2}^{n}\frac{1}{\lambda_i^2(G)}, $ where $ \lambda_i(G) $ is the $ i $-th eigenvalue of the Laplacian matrix of $ G $ with $ n $ vertices. In this paper, we provide the mathematical relationships between the biharmonic index and some classic topological indices: the first Zagreb index, the forgotten topological index and the Kirchhoff index. In addition, the extremal value on the biharmonic index for all graphs with diameter two, trees and firefly graphs are given, respectively. Finally, some graph operations on the biharmonic index are presented.

Multiplicative degree-Kirchhoff index is a very interesting topological index. In this article, we compute analytical expression for the expected value of the Multiplicative degree-Kirchhoff index in a random polygonal. Based on the result above, we also get the Multiplicative degree-Kirchhoff index of all polygonal chains with extremal values and average values. [ABSTRACT FROM AUTHOR]

The additive degree–Kirchhoff index is an important topological index. This paper we devote to establishing the explicit analytical expression for the simple formulae of the expected value of the additive degree–Kirchhoff index in a random polygon. Based on the result above, the additive degree–Kirchhoff indexes of all polygonal chains with extremal values and average values are obtained. [ABSTRACT FROM AUTHOR]

In organic chemistry (especially in polycyclic aromatic compounds), hexagonal and quadrilateral molecular structures are very common. Let P h be the set of phenyenes with h hexagons and h − 1 quadrilaterals. The Mostar index Mo(G) is defined as M o (G) = ∑ e = u v ∈ E (G) | n u − n v | , where nu (resp., nv) is the number of vertices whose distance to vertex u (resp., v) is smaller than the distance to vertex v (resp., u). In this paper, we completely determine the maximal values of the Mostar index of tree-type phenylenes with one full-hexagon and characterize all the tree-type phenylenes attaining these values. Moreover, we give some properties of tree-type phenylenes with maximal Mostar index. [ABSTRACT FROM AUTHOR]

The Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph G as S O (G) = ∑ u v ∈ E (G) d u 2 + d v 2 , where d u = d G (u) denotes the degree of vertex u in graph G. Extremal problems on the Sombor index for trees with a given diameter has been considered by Chen et al. (MATCH Commun Math Comput Chem 87:23–49, 2022) and Li et al. (Appl Math Comput 416:126731, 2022). As an extension of the results introduced above, we determine the maximum Sombor indices for unicyclic graphs with a fixed order and given diameter. [ABSTRACT FROM AUTHOR]

Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.

Recently the exponential Randić index $ {{\rm e}^{\chi}} $ was introduced. The exponential Randić index of a graph $ G $ is defined as the sum of the weights $ {{\rm e}^{{\frac {1}{\sqrt {d \left(u \right) d \left(v \right) }}}}} $ of all edges $ uv $ of $ G $, where $ d(u) $ denotes the degree of a vertex $ u $ in $ G $. In this paper, we give sharp lower and upper bounds on the exponential Randić index of unicyclic graphs.

The Sombor index was introduced in 2020, and was soon followed by a remarkably large number of studies, both chemical and mathematical. In this review we collect the existing bounds and extremal results related to the Sombor index and its variants. [ABSTRACT FROM AUTHOR]

A vertex-degree-based (VDB, for short) topological index φ is defined for a graph G with n vertices as φ (G) = ∑ 1 ≤ i ≤ j ≤ n − 1 m i , j (G) φ (i , j) , where m i , j = m i , j (G) is the number of edges of G joining a vertex of degree i with a vertex of degree j, and { φ (i , j) } is a set of real numbers. In this paper, we present a general formula to calculate VDB topological indices of graphene systems. As a byproduct, we solve the following extremal value problem: given a VDB topological index φ , find the graphene with maximal value of φ and the graphene with minimal value of φ , among all graphene with h hexagons. [ABSTRACT FROM AUTHOR]

For a connected graph G , the Mostar index is defined as Mo (G) = ∑ x y ∈ E (G) | n x − n y | , where n x (respectively, n y ) is the number of vertices of G closer to x (respectively, y) than y (respectively, x). In this paper, we determine the first five maximal (respectively, the first five minimal) values of the Mostar index among all phenylene chains with h hexagons, the corresponding extremal chains are completely characterized, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate general Lp-mixed brightness integrals which were defined by Yan and Wang. We give their extremal values and establish Brunn-Minkowski type inequalities. [ABSTRACT FROM AUTHOR]

Computing topological indices of molecular structures is a fundamental and classical topic. In organic chemistry, hexagonal and quadrilateral molecular structures are very common. The detour index ω (G) of a connected graph G is defined as ω (G) = ∑ { u , v } ⊆ V (G) l G (u , v) , where l G (u , v) denotes the detour distance between vertices u and v. In this study, we obtain the explicit analytical expression for detour index of phenylene chains with a fixed number of hexagons. Further minimal and maximal phenylene chains are obtained. [ABSTRACT FROM AUTHOR]

Multiplicative degree-Kirchhoff index is a very interesting topological index. In this article, we compute analytical expression for the expected value of the Multiplicative degree-Kirchhoff index in a random polygonal. Based on the result above, we also get the Multiplicative degree-Kirchhoff index of all polygonal chains with extremal values and average values.

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Based on elementary geometry, a class of novel graph invariants was introduced by Gutman, of which the simplest is the Sombor index SO, defined as SOG=∑uv∈EdG2u+dG2v, where G = (V, E) is a simple graph and dG(v) denotes the degree of v in G. In this paper, the chemical importance of the Sombor index is investigated and it is shown that the new index is useful in predicting physico‐chemical properties with high accuracy compared to some well‐established and often used indices. We obtain a sharp upper bound for the Sombor index among all molecular trees with fixed numbers of vertices, and characterize those molecular trees achieving the extremal value. Also, we obtain the extremal values of the reduced Sombor index for molecular trees. [ABSTRACT FROM AUTHOR]

Wang and Liu proposed the definition of log radial sum and researched the dual log-Brunn-Minkowski theory. Motivated by the log radial sum of star bodies, we define the general log radial bodies and explore some related properties. For this notion and its the star dual form, we prove two extremum inequalities for dual quermassintegrals. A star dual log-Minkowski inequality is established by us. [ABSTRACT FROM AUTHOR]

The additive degree–Kirchhoff index is an important topological index. This paper we devote to establishing the explicit analytical expression for the simple formulae of the expected value of the additive degree–Kirchhoff index in a random polygon. Based on the result above, the additive degree–Kirchhoff indexes of all polygonal chains with extremal values and average values are obtained.

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation −Δu + V (x)u = λa(x)u−γ + b(x)upin ℝN, beyond the Nehari extremal value, as defined in [Y. Il’yasov, On extreme values of Nehari manifold via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017) 683–714], when the potential b ∈ L∞(ℝN) may change its sign, 0 < a ∈ L 2 1+γ(ℝN), V is a positive continuous function, N ≥ 3 and λ > 0 is a real parameter. The main difficulties come from the non-differentiability of the energy functional and the fact that the intersection of the boundaries of the connected components of the Nehari set is non-empty. We overcome these difficulties by exploring topological structures of that boundary to build non-empty sets whose boundaries have empty intersection and minimizing over them by controlling the energy level. [ABSTRACT FROM AUTHOR]

An extremal value of a function is the value of the function at one of its extremum points. Each extremum point of a differentiable function of several variables is described by the system of equations expressing the vanishing of all partial derivatives. However, in the general case, one cannot obtain equations for extremal values of the function. The case of polynomials significantly differs from the general case, and in this paper we obtain an equation for extremal values of a given polynomial in several variables. [ABSTRACT FROM AUTHOR]

Unicyclic graph is one kind of typical graph for indices basing degree of vertex. Various chemical structures can be exhibited in unicyclic graph as well. A graph is said to be unicyclic if the graph is connected and | V (G) | = | E (G) |. Recently, the Sombor index, which is defined by S O = S O (G) = ∑ u v ∈ E (G) d G 2 (u) + d G 2 (v) , was proposed by Gutman. This paper establishes distinct bounds for this index of unicyclic graph with girth l , as well as specific bounds of chemical unicyclic graph with girth l. • Some bounds and extremal graphs of Sombor index over unicyclic graph are given. • The minimum value and graph of Sombor index over chemical unicyclic graph are gotten. • With a order condition, the maximum value and graphs of Sombor index over chemical unicyclic graph are determined. [ABSTRACT FROM AUTHOR]

A vertex-degree-based (VDB, for short) topological index φ induced by the numbers φij was recently defined for a digraph D, as φD=12∑uvφdu+dv−, where du+ denotes the out-degree of the vertex u,dv− denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over Dn, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.

Based on the notion of Lp-mean zonoids which is introduced by Xi, Guo and Leng, we define the notion of asymmetric Lp-mean zonoids and establish its an affine inequality. Further, we give the extremal values for volume, quermassintegrals of this notion and dual quermassintegrals of polar of this notion. Finally, associated with p-radial mean bodies, two monotonic inequalities for the asymmetric Lp-mean zonoids are obtained. [ABSTRACT FROM AUTHOR]

Recently, based on elementary geometry, Gutman proposed several geometry-based invariants (i.e., SO , S O 1 , S O 2 , S O 3 , S O 4 , S O 5 , S O 6). The Sombor index was defined as S O (G) = ∑ u v ∈ E (G) d u 2 + d v 2 , the first Sombor index was defined as S O 1 (G) = 1 2 ∑ u v ∈ E (G) | d u 2 − d v 2 | , where d u denotes the degree of vertex u. In this paper, we consider the mathematical properties of these geometry-based invariants. We determine the maximum trees (resp. unicyclic graphs) with given diameter, the maximum trees with given matching number, the maximum trees with given pendent vertices, the maximum trees (resp. minimum trees) with given branching number, the minimum trees with given maximum degree and second maximum degree, the minimum unicyclic graphs with given maximum degree and girth, the minimum connected graphs with given maximum degree and pendent vertices, and some properties of maximum connected graphs with given pendent vertices with respect to the first Sombor index S O 1. As an application, we inaugurate these geometry-based invariants and verify their chemical applicability. [ABSTRACT FROM AUTHOR]

Shannon entropies for networks have been widely introduced. However, entropies for weighted graphs have been little investigated. Inspired by the work due to Eagle et al., we introduce the concept of graph entropy for special weighted graphs. Furthermore, we prove extremal properties by using elementary methods of classes of weighted graphs, and in particular, the one due to Bollobás and Erdös, which is also called the Randi´c weight. As a result, we derived statements on dendrimers that have been proven useful for applications. Finally, some open problems are presented.

The Wiener polarity index (usually denoted by W p ) of a graph G is defined as the number of unordered pairs of the vertices of G which are at distance 3. Denote by CT n the family of all n -vertex chemical trees. In a recent paper, Ashrafi and Ghalavand [1] determined the first three minimum W p values of n -vertex chemical trees for n  ≥ 7 and characterized the chemical trees attaining the first two minimum W p values among all the members of CT n for n  ≥ 4. In this note, the chemical trees with the third minimum W p value are characterized from the graph family CT n for n  ≥ 7, and the chemical trees from the family CT n , n  ≥ 4, with the first two minimum W p values are also obtained in an alternative but shorter way. [ABSTRACT FROM AUTHOR]

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2/di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of n-vertex trees with two adjacent vertices of maximum degree Δ, where n ≥ 2Δ. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randićc index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas [11]. [ABSTRACT FROM AUTHOR]

In this article, based on the Blaschke combination of convex bodies, we define the general Blaschke bodies and obtain the extremal values of their volume and affine surface area. Further, we study the asymmetric negative solutions of the Shephard problem for the projection bodies.

We consider the singular boundary value problem ... where Ω ⊂ RN(N ≥ 3) is a bounded domain such that 0 ∊ Ω, 0 < γ < 1, 0 ≤ a < N-p/p, a ≤ b < a + 1, p* := p*(a, b) = Np/N-(1+a-b)p, and h(x) is a given function. Based on different assumptions, using variational methods and Ekeland's principle, we admit that this problem possesses two positive solutions. [ABSTRACT FROM AUTHOR]

We consider the singular boundary value problem { − Δ u − λ V ( x ) u = h ( x ) u − γ + μ u 2 ⁎ − 1 in Ω , u ( x ) > 0 in Ω , u ( x ) = 0 on ∂ Ω , where 0 < λ < λ ¯ = ( N − 2 2 ) 2 , 0 < γ < 1 , 2 ⁎ = 2 N N − 2 , h ( x ) is a given function and V ( x ) has prescribed finitely many singular points. Our goal in this paper is to establish some existence and multiplicity results for above problem when μ ∈ ( 0 , μ ⁎ ) for some μ ⁎ > 0 and obtain exact estimate for extremal value μ ⁎ = μ ⁎ ( Ω , γ , 2 ⁎ , h ( x ) ) > 0 for above problem. [ABSTRACT FROM AUTHOR]

Lutwak defined the dual Blaschke combination of star bodies. In this paper, based on the $L_{p}$-dual Blaschke combination of star bodies, we define the general $L_{p}$-dual Blaschke bodies and obtain the extremal values of their volume and $L_{p}$-dual affine surface area. Further, as the applications, we study two negative forms of the $L_{p}$-Busemann-Petty problems. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that solutions of ordinary differential equations are continuously dependent on finite-dimensional parameters in equations. In this paper we study the dependence of solutions and eigenvalues of second-order linear measure differential equations on measures as an infinitely dimensional parameter. We will provide two fundamental results, which are the continuity and continuous Fréchet differentiability in measures when the weak⁎ topology and the norm topology of total variations for measures are considered respectively. In some sense the continuity result obtained in this paper is the strongest one. As an application, we will give a natural, simple explanation to extremal problems of eigenvalues of Sturm–Liouville operators with integrable potentials. [Copyright &y& Elsevier]

The Gutman index and Schultz index of a connected graph are degree-distance-based topological indices. In this paper, we devoted to establish the explicit analytical expressions for the simple formulae of the expected values of the Gutman and Schultz indices in a random polygonal. Based on these results above, we get the extremal values and average values of Gunman and Schultz indices of all polygonal chains.