The spectral graph theory investigates the relationships between combinatorial qualities of graphs and algebraic properties of related matrices. The adjacency matrix is currently undergoing significant modification as a result of its well-developed theoretical and application standpoint. The present work deals with one such extension of the adjacency matrix. We propose here the neighborhood Sombor matrix corresponding to the well-known Sombor index. We compute the neighborhood Sombor spectrum of some benchmark graphs. Lower and upper bounds of the spectral radius (zeta 1) are derived with identifying extremal graphs. Moreover, extremal trees are characterized in view of spectral radius, where path and star graphs yield minimal and maximal structures, respectively. The role of zeta 1 in structure-property modelling is also demonstrated. The isomer-discrimination ability of zeta 1 is found to be better than that of some well-known descriptors.

The study of topological descriptors is essential for understanding the underlying structures of graphs and networks. Numerous numerical descriptors associated with graphs have been used to analyze their overall structure. In this analysis, degree-based topological descriptors hold a significant place. The Euler Sombor index of a graph Gamma is a significant vertex degree-based topological index related to the Sombor index, showing a strong correlation with the physicochemical properties of octanes. It represents the perimeter of an ellipse, with focal points recognized as the degree-point and the dual-point of a pair of connected vertices in Gamma. Nowadays, finding extremal results with respect to various graph indices for fixed graph parameters has become an important and engaging area of research in extremal graph theory. In this article, we explore the first and second maximum Euler Sombor index of a tree with a fixed diameter d >= 4. The extremal trees are also identified. Furthermore, we provide the ordering of trees for d=3. In addition, we identify the maximal unicyclic graph when the graph order and diameter are given.

The study of graph complexity has led to a deeper understanding of the structures of graphs. This paper presents new findings on the Szeged complexity of graphs. Specifically, we prove that for bipartite graphs on n vertices, the upper bound of Szeged complexity is ⌊n/2⌋ . Moreover, we establish that the lower bound of Szeged complexity of a tree T is the radius of T. Furthermore, we characterize trees with Szeged complexity three and determine their Wiener complexity.

State transfer has great significance due to its important applications in quantum information processing and quantum computing. The ability to transfer quantum states efficiently is critical for the development of quantum networks and various quantum algorithms. In this paper, we delve into the exploration of pretty good state transfer on Cayley graphs over U_6n for n = 2^r, n = 2^rp^t and n = mp (where m can be either odd or even and p is an odd prime number).

Fullerenes are unique carbon-cage molecules characterized by a nearly spherical arrangement of carbon atoms. This study introduces the concept of the symmetric division eccentric index (SDE) of a graph which quantifies the relationships between vertices based on their eccentricities. We explore various bounds for this index specifically in the context of fullerene graphs. Additionally, we calculate the SDE for two infinite classes of fullerenes providing insights into their topological properties. Our analysis reveals a strong correlation (correlation coefficient of 0.98) between the sde-index and the boiling points of various fullerenes suggesting that this new topological index may serve as a predictive tool for estimating thermal properties.

Let G be a connected graph. The generalized distance matrix is defined as Tr alpha(G) = alpha Tr(G) + (1 - alpha)D(G), where 0 <= alpha <= 1, D(G) is the distance matrix whose (i,j)th entry is the distance between the ith and jth vertices in G and Tr(G) is the diagonal matrix where each nonzero entry corresponds to the total distance to a vertex. The generalized distance spread, STr alpha(G), was defined as the difference between the largest eigenvalue and the smallest eigenvalue of Tr alpha(G). In this paper some lower bounds for STr alpha(G) in terms of alpha, the Wiener index and n is presented and the equality cases are discussed. For a graph with prescribed diameter, a lower bound for STr alpha(G) as function of the vertex transmissions, distances between vertices, the diameter and alpha is presented. A lower bound for this graph invariant in the class of trees is given. Additionally, lower bounds for STr alpha(G) as function of some known topological indices are presented. Some closed formulas for STr alpha(G) of known vertex-transitive graphs such as the polyhex nanotorus and hypercube are presented as applications of previous results. Moreover, some lower bounds (as function of alpha) for some known molecular graphs are also presented.

This paper introduces a novel graph invariant, the symmetric division Szeged index (SDZ), which generalizes earlier concepts by focusing on vertices positioned closer to an edge's endpoints rather than vertex degrees. It explores several properties and inequalities associated with the SDZ-index, offering examples and results for various graph classes, including bipartite graphs, trees, complete graphs, unicycles, distance-balanced graphs, and triangle-free graphs. Additionally, the SDZ-index is contrasted with other well-known graph indices. The behavior of the SDZ-index under graph operations, such as corona, sum, lexicographic, and Cartesian products, is also examined. The study highlights the index's potential in topological analysis, particularly in revealing statistically significant correlations with the molecular properties of octane isomers. As research progresses, we anticipate further developments and applications that will enhance our comprehension of complex systems and their properties across fields like network science, computer science, and physics.

A fullerene graph is a 3-connected, planar cubic graph where each face is either pentagonal or hexagonal. This paper investigates the symmetric division eccentric index (SDE) of fullerenes, particularly focusing on two infinite classes F-10n and F-12n. We establish several bounds for fullerene graphs. We also explore the automorphism group actions on the vertices and edges of these fullerene graphs, establishing a relation between edge orbits and their eccentricities. General formulas for calculating SDE-indices are derived for F-10n, when n >= 8 and F-12n, when n >= 10. Furthermore, we present a new approach to compute the SDE-index and then implement the method to obtain general formulas for the SDE-index of the given classes of fullerenes. (c) 2025 University of Kashan Press. All rights reserved.

For a graph G with orbits O1,…,Ok, the orbit polynomial is defined as OG(x)=∑i=1kx|Oi|. In the current work, we will characterize all fullerenes concerning their pentagonal orbit polynomials (namely, the orbit polynomials for orbits of pentagons). Besides, the orbit polynomial, the orbit-entropy, the symmetry index and the unique positive root of orbit polynomial are computed for several infinite families of fullerenes. Finally, correlations between these graph measures are established to quantify their relationship.

Growing up in South Tehran, Ali Reza Ashrafi had a fascination with the application of mathematics in other sciences, particularly the symmetry of molecules, despite having no formal training in the subject. Ashrafi was a prominent mathematician who made significant contributions to the study of symmetry in molecular graphs. His work on this subject has had a profound impact on the field of chemistry, helping chemists to better understand the structure and properties of molecules. With great enthusiasm, Ashrafi explains to his students that the symmetry group plays a significant role in everything and that symmetry can be used to predict or explain many of a molecule's chemical properties.

The cozero divisor graph Gamma'(R) of a commutative ring R is a simple graph with vertex set as non-zero zero divisor elements of R such that two distinct vertices x and y are adjacent iff x is an element of/ Ry and y is an element of/ Rx, where xR is the ideal generated by x. In this article we find the spectra of Gamma'(Zn) for n is an element of {q1q2, q1q2q3, qn1 primes. As a consequence we obtain the bounds for the largest (smallest) eigenvalues, bounds for spread, rank and inertia of Gamma'(Zqn1 1 q2) along with the determinant, inverse and square of trace of its quotient matrix. We present the extremal bounds for the energy of Gamma'(Zn) for n = qn1 1 q2 and characterize the extremal graphs attaining them. We close article with conclusion for furtherance.

In this article, we survey the results on examining orbit structures combined with polynomials, automorphism groups, roots of polynomials, and the construction of graphs with prescribed orbit structures. The orbit polynomial has been defined as the Pncxn, wherecrepresents the number of orbits of graph G with sizen. By subtracting this polynomial from 1, the modified orbit polynomial O star G(x) = 1-OG(x) is obtained which possesses a unique positiveroot denoted by delta. This root can be seen as a relative measure ofsymmetry. The study of orbit structures in graphs and their associ-ated automorphism groups is a fundamental topic in graph theory.The understanding and analysis of these structures provide valuableinsights into the symmetries in graphs, enabling the exploration ofvarious graph properties and their applications in diverse fields such as network analysis, computer science, and chemistry

Following the symmetric division deg index, this article introduces a new graph index called the SDE index which is based on the eccentrity of vertices. Also, the eccentric version of the extended adjacency matrix is defined. We establish several results regarding the properties of the extended eccentric matrix and spectra of different classes of graphs including trees and unicyclic graphs. Some new bounds also for the minimum spectral radius of the extended eccentric matrices were determined. Besides, we present bounds for the extended eccentric spectral radius of path graph and prove that the maximum value of the extended eccentric spectral radius in a tree is attained by the star graph. In our research, we explore the potential applications of the SDE index in chemical graph theory. Our findings illustrate a noteworthy correlation between the SDE index and several physical and chemical properties. This correlation highlights the promising potential of the index as a predictive tool for the molecular behavior of various compounds. For example, the SDE index has a stronger negative correlation with the M1, M2, and M3 indices (r = -0.93, -0.94, and -0.91, respectively).

The study delves into the relationship between symmetry groups and automorphism groups in polyhedral graphs, emphasizing their interconnected nature and their significance in understanding the symmetries and structural properties of fullerenes. It highlights the visual importance of symmetry and its applications in architecture, as well as the mathematical structure of the automorphism group, which captures all of the symmetries of a graph. The paper also discusses the significance of groups in Abstract Algebra and their relevance to understanding the behavior of mathematical systems. Overall, the findings offer an inclusive understanding of the relationship between symmetry groups and automorphism groups, paving the way for further research in this area.

The definition of the weighted topological index associated with a degree function phi is Phi(G) = Euv is an element of E (G) phi(du,dv), where du denotes the degree of node u and phi satisfies symmetric property phi(du, dv) = phi(dv, du). In this paper, we characterized extremal graphs and presented several results concerning the function Phi(G) in terms of various graph invariants. Additionally, we characterize the graphs that achieve these bounds and present multiple bounds for Phi(G) for the class of cozero divisor graphs defined on commutative rings.

In this article, we present an enhanced version of the symmetric division deg index (sdd-index) known as symmetric division eccentric index or SDE index, for short. Unlike its predecessor, SDE employs eccentricity instead of vertex degree to assess the properties of a graph G. In this paper, we first give some bounds for SDE index of a connected graph G with fixed size m. For two connected graphs G_1 and G_2 of order n_1 and n_2 , employing these bounds, we compute the SDE index for two classes of graph products, e.g., the Cartesian product and Corona product. As an application, we determine the structure of graphs with two non-equi-centric edges. Our theorems generalize the recent results for the extended adjacency index of a graph. Besides, this research significantly contributes to the comprehension of graph analysis techniques and offers valuable insights into the relationship between SDE and various graph properties.

When characterizing networks structurally, the discriminating ability of a topological index is crucial. This relates to investigate its discrimination power (also called uniqueness or degeneracy) that indicates how meaningful the given measure can distinguish nonisomorphic networks. Assume G is a connected graph. The Szeged complexity (or briefly Sz-complexity) of a graph G is the number of different portions in the Szeged index formula. Also, the Wiener complexity (or briefly W-complexity) can be defined similarly. In the current work, we study graphs with small Sz-complexity. We characterize trees with Sz-complexity two and bicyclic graphs with Sz-complexity one. In this way, first we introduce some graphs with Sz-complexity one. For instance, we investigate Θ-graph and two categories of k-cyclic graphs. Besides, we classify bicyclic graphs with Sz-complexity equal to the number of edge-orbits. Finally, we determine W-complexity of these graphs.

In this article, we introduce a new counting polynomial, namely the orbit polynomial. It is well-known that this polynomial has a unique positive zero δ in the interval [0, 1]. The aim of this paper is to study the specific properties of this polynomial and then determine the location of this root for several classes of complex networks to compare with other graphical measures. Additionally, we compare the unique positive zero measure with several well-known centrality graph measures.

Suppose G is a connected graph. The Wiener complexity (or briefly W-complexity) CW(G) is the number of different contributions to Wiener index in its summation formula. Also, the Szeged complexity (or briefly Sz-complexity) CSz(G) can be defined similarly. The main goal of the current work is to investigate graphs with small complexity. We classify all unicycle graphs, all distance-balanced graphs, and all regular graphs of diameter 2 with Sz-complexity one. In this way, the Sz-complexity and W-complexity of several families of graphs are determined.