Let $G$ be a simple undirected graph with vertex set $V(G)=\{v_1, v_2,\ldots,v_n\}$ and edge set $E(G)$. The Sombor matrix $\mathcal{S}(G)$ of a graph $G$ is defined so that its $(i,j)$-entry is ...equal to $\sqrt{d_i^2+d_j^2}$ if the vertices $v_i$ and $v_j$ are adjacent, and zero otherwise, where $d_i$ denotes the degree of vertex $v_i$ in $G$. In this paper, lower and upper bounds on the spectral radius, energy and Estrada index of the Sombor matrix of graphs are obtained, and the respective extremal graphs are characterized.
Let G be a graph with vertex set V and edge set E. A topological index has the formTI(G)=∑uv∈Ef(du,dv), where f=f(x,y) is a pertinently chosen function which must be symmetric and real-valued for all ...x,y pertaining to vertex degrees of the graph G. Particularly interesting are the Sombor index SO and the elliptic Sombor index ESO, induced by the functions f(x,y)=x2+y2 and f(x,y)=(x+y)x2+y2, respectively. In this paper we analyze the ordering relations in benzenoid systems with respect to these two important topological indices. Also, we extend the results to general Sombor index SOα,β and general elliptic Sombor index ESOα.
On integral Sombor indices Rada, Juan; Rodríguez, José M.; Sigarreta, José M.
Applied mathematics and computation,
09/2023, Letnik:
452
Journal Article
Recenzirano
Let G=(V(G),E(G)) be a simple graph and denote by du the degree of the vertex u∈V(G). Using a geometric approach, I. Gutman introduced a new vertex-degree-based topological index, defined ...asSO(G)=∑uv∈E(G)(du)2+(dv)2,and named Sombor index. It is a molecular descriptor with an impressive research activity in recent years. In this paper we propose and initiate the study of a family of topological indices, also conceived from a geometrical point of view, called integral Sombor indices, that generalize the Sombor index. Also, we study the application of these indices in QSPR/QSAR research.
A chemical graph is a model used to indicate a chemical combination. In a molecular graph, vertices define atoms, and edges are represented as chemical bonds. A topological index is a single number ...to characterize the graph of a molecule. In this article, we study the topological properties of some special chains. The polyphenyl chains with hexagons are graphs of aromatic organic compounds. The key purpose of this article is to explore the expected value of Sombor, reduced Sombor, and average Sombor index for this category of organic compounds. It was investigated that the Sombor, reduced Sombor and average Sombor index revealed adequate discriminative potential of alkanes. It has been tested that these indicators can be used effectively in modeling chemical thermodynamic structures. The average value of the Sombor, reduced Sombor, and average Sombor index for the set of all spiro and random polyphenyl chains has been determined. Finally, the ratio between the expected values of these mentioned indices for both random chains has been resolved. Doi: 10.28991/ESJ-2022-06-01-012 Full Text: PDF
The reduced Sombor index SOred(G) and the exponential reduced Sombor index eSOred(G) of a graph G are defined respectively ...asSOred(G)=∑uv∈E(G)(d(u)−1)2+(d(v)−1)2,eSOred(G)=∑uv∈E(G)e(d(u)−1)2+(d(v)−1)2, where d(u) denotes the degree of the vertex u in G. In this paper, we obtain the maximum value of the reduced Sombor index among all molecular trees of order n with perfect matching and characterize the corresponding extremal trees. This solves a problem of the reduced Sombor index posed by Deng, Tang and Wu (2021) 5. We also show that the maximum molecular trees of exponential reduced Sombor index and reduced Sombor index are the same, which was conjectured by Liu, You, Tang and Liu (2021) 9.
Let G be a simple graph with the vertex set V={v1,…,vn} and denote by dvi the degree of the vertex vi. The modified Sombor index of G is the addition of the numbers (dvi2+dvj2)−1/2 over all of the ...edges vivj of G. The modified Sombor matrix AMS(G) of G is the n by n matrix such that its (i,j)-entry is equal to (dvi2+dvj2)−1/2 when vi and vj are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of AMS(G). The sum of the absolute eigenvalues of AMS(G) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is 2; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven.
In this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain COCn. Meanwhile, the explicit formulas of ...the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the expected values of the Sombor index SOCOCn, the reduced Sombor index RSOCOCn, the modified Sombor index mSOCOCn and the reduced and modified Sombor index mRSOCOCn for a random cyclooctane chain.
Introduction/purpose: The Euler-Sombor index (EU) is a new vertexdegree-based graph invariant, obtained by geometric consideration. It is closely related to the Sombor index (SO). The actual form of ...this relation is established. Methods: Combinatorial graph theory is applied. Results: The inequalities between EU and SO are established. Conclusion: The paper contributes to the theory of Sombor-index-like graph invariants.
For a simple connected graph G=(V,E), let d(u) be the degree of the vertex u of G. The general Sombor index of G is defined as SOα(G)=∑uv∈Ed(u)2+d(v)2α where SO(G)=SO0.5(G) is the recently invented ...Sombor index. In this paper, we show that in the class of connected graphs with a fixed degree sequence (for which the minimum degree being equal to one), there exists a special extremal BFS-graph with minimum general Sombor index for 0<α<1 (resp. maximum general Sombor index for either α>1 or α<0). Moreover, for any given tree, unicyclic, and bicyclic degree sequences with minimum degree 1, there exists a unique extremal BFS-graph with minimum general Sombor index for 0<α<1 and maximum general Sombor index for either α>1 or α<0.
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