The v-irregularity, a variant of the well-established Albertson irregularity, is a topological invariant defined for a graph G = (V, E) as v(G) = & sum;uv is an element of E(d(u) - d(v))2, where d(u) and d(v) denote the degrees of vertices u and v, respectively. Recent research has successfully characterized chemical trees with the maximum v-irregularity. In this paper, we expand upon this research by establishing several structural properties of maximal trees with prescribed maximum degree triangle. Application of these properties enables us to characterize maximal trees with triangle = 5. We establish that extremal trees contain only vertices of degrees 1, 2 and triangle. Moreover, the number of edges with both end-vertices having the degree 2 or triangle is very small, so almost all edges have the (second) maximum possible contribution to v-irregularity. We believe this property or similar should extend to maximal trees for any value of triangle, so this is an interesting direction for further research.

Let G be a graph and denote its vertex set and edge set by V(G) and E(G), respectively. For a vertex v(i) is an element of V(G), let d(i) denote its degree. A broad class of numerical parameters for BIDv(G) = Sigma v(i)v(j)(is an element of E(G)) v(d(i), d(j)), where the function v is symmetric and it assigns a real value to each pair of degrees of adjacent vertices of G. Such graph parameters are known as bond incident degree (BID) indices. The family BIDv specializes to the general atom-bond connectivity index ABC(alpha) when v(d(i), d(j)) = ((d(i), d(j )- 2)d(i)(-1) d(j)(-1))(alpha), for any real parameter alpha; in particular, the choice alpha = 1/2 yields the classical atom-bond connectivity index. In their work (Chen and Hao, 2018), Chen and Hao posed the problem of identifying those graphs from the class of all connected graphs of fixed order with prescribed edge or vertex connectivity that attain the maximum value of ABC(alpha) for any alpha with 0 < alpha < 1/2. The present article resolves the aforementioned problem by providing a general result for BID & vartheta; under some suitable conditions imposed on v. These conditions are fulfilled not only by ABC(alpha) for 0 < alpha < 1/2, but also by many other existing particular BID indices, such as the reformulated first Zagreb index, the Sombor index and its reduced form, the Euler-Sombor index, the inverse sum indeg index, the Zagreb-Sombor index, the reciprocal sum-connectivity index, the reciprocal Randi & cacute; index, and the elliptic Sombor index. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A (molecular) graph in which all vertices have the same degree is known as a regular graph. According to Gutman, Hansen, and Mélot [J. Chem. Inf. Model. 45 (2005) 222-230], it is of interest to measure the irregularity of nonregular molecular graphs both for descriptive purposes and for QSAR/QSPR studies. The graph invariants that can be used to measure the irregularity of graphs are referred to as irregularity measures. One of the well-studied irregularity measures is the “total irregularity” measure, which was introduced about a decade ago. Bounds and optimization problems for this measure have already been extensively studied. A considerable number of existing results (concerning this measure) also hold for molecular graphs; particularly, the ones regarding lower bounds and minimum values of the mentioned measure. The primary objective of the present review article is to collect the existing bounds and optimal results concerning the total irregularity measure. Several open problems related to the existing results on the total irregularity measure are also given.

The σ_t-irregularity (or sigma total index) is a graph invariant which is defined as σ_t(G)=∑_{u,v}⊆ V(G)(d(u)-d(v))^2, where d(z) denotes the degree of z. This irregularity measure was proposed by Réti [Appl. Math. Comput. 344-345 (2019) 107-115], and recently rediscovered by Dimitrov and Stevanović [Appl. Math. Comput. 441 (2023) 127709]. In this paper we remark that σ_t(G)=n^2· Var(G), where Var(G) is the degree variance of the graph. Based on this observation, we characterize irregular graphs with maximum σ_t-irregularity. We show that among all connected graphs on n vertices, the split graphs S_⌈n/4⌉, ⌊3n/4⌋ and S_⌊n/4⌋, ⌈3n/4⌉ have the maximum σ_t-irregularity, and among all complete bipartite graphs on n vertices, either the complete bipartite graph K_⌊n/4(2-√(2))⌋, ⌈n/4(2+√(2))⌉ or K_⌈n/4(2-√(2))⌉, ⌊n/4(2+√(2))⌋ has the maximum sigma total index. Moreover, various upper and lower bounds for σ_t-irregularity are provided; in this direction we give a relation between the graph energy ℰ(G) and sigma total index σ_t(G) and give another proof of two results by Dimitrov and Stevanović. Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between σ_t and σ. We conclude the paper with two conjectures.

Let G be a graph with edge set E(G). Let dx denote the degree of a vertex x in G. For a nonnegative integer k, a connected graph of order n and size n + k-1 is called a kcyclic graph. This paper is concerned with k-cyclic graphs and their graphical indices of the form BIDf(G) = Sigma uv is an element of E(G) f(du, dv), where f is a symmetric function whose outputs are real numbers. Particularly, the graphs minimizing or maximizing BIDf among all k-cyclic graphs with a given order are studied under certain constraints on f. Various existing indices meet these constraints, and hence the obtained results hold for those indices; more precisely, one of the obtained results covers the recently developed elliptic Sombor and Zagreb-Sombor indices, while another result covers the recently introduced Euler-Sombor index.

The total sigma-irregularity is given by sigma(t)(G)=& sum;({u,v}subset of V(G))(d(G)(u)-d(G)(v))(2), where d(G)(z) indicates the degree of a vertex z within the graph G. It is known that the graphs maximizing sigma(t)-irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to sigma(f(n))(t)(G)=& sum;({u,v}subset of V(G))|d(G)(u)-d(G)(v)|(f(n)), where n=|V(G)| and f(n)>0. We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.

Irregularity measures of graphs serve as crucial tools for optimizing networks, understanding biological interactions, analyzing social dynamics, enhancing cybersecurity, and assessing market stability, offering valuable insights across diverse fields. In this paper, we introduce a family of irregularity measures for graphs with non-increasing degree sequences, termed degree scaling irregularities, defined as ℐ(G,r)=∑ _i=1^n d_i r_i , where d=(d_1,d_2, … ,d_n) is a non-increasing degree sequence of a graph G and r = (r_1, r_2, … , r_n) is a non-increasing n-tuple of real numbers such that r_1+r_2+ ⋯ + r_n=0 . This family provides a versatile framework for analyzing graph irregularity. Various choices for r are explored, including using eigenvalues of matrices related to G or orientations of G. It has been proven that if G maximizes ℐ -irregularity among all connected graphs of order n for a given r, then G is a split graph, a graph comprised of a clique and an independent set. Additionally, we investigate the properties of graphs that maximize ℐ -irregularity and explore computational aspects when r is based on matrix eigenvalues.

Let G(alpha) be the graph obtained from a simple graph G of order n by adding sigma self-loops, one self-loop at each vertex in S subset of V (G). Let lambda(1)(G(sigma)), lambda(2)(G(sigma)),...,lambda(n)(G(sigma)) be the eigenvalues of G(sigma). The energy of G(sigma), denoted by E (G(sigma)), is defined as E (G(sigma)) = (i=1)Sigma(n) vertical bar lambda(i)(G(sigma)) -sigma/n vertical bar. In this paper, using various analytic inequalities and previously established results, we derive several new lower and upper bounds on E(G(sigma)).

This article gives bounds on a substantial number of BID (bond incident degree) indices for connected graphs in terms of their order, size, and maximum degree. The considered BID indices include, among others, the Sombor index (together with its reduced version), atom-bond sum-connectivity index, symmetric division deg index, sum-connectivity index, harmonic index, and Randi´c index. All the graphs that attain the obtained bounds are also characterized. All the established bounds are valid also for molecular graphs. A graph of order n and size m is called an (n,m)-graph. The obtained bounds provide a partial solution to the problem of finding graphs with extremum (considered) BID indices over the class of all connected (n,m)-graphs with a fixed maximum degree under certain constraints.

The problem of complete characterization of trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most challenging and intriguing open problems in mathematical chemistry. Recently, the problem has been completely solved. Here, we provide an overview of the key results that led to its complete solution.

The s-irregularity index is a variant of the well-established Albertson irregularity index. For a graph G = (V, E) it is defined as sigma(G) = Sigma(uv is an element of E) (d(u) - d(v))(2), where d(u) and d(v) denote the degrees of vertices u and v, respectively. In this note, we characterize chemical trees of a given order with maximal sigma-irregularity index.

The problem of complete characterization of trees with minimal atom-bond-connectivity index (minimal-ABC trees) is a long-standing open problem. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called Dz- and Dz+1-branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order.

Abstract The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph G G of order n n and size m m , it is defined as CS ( G ) = λ 1 − 2 m / n , \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)={\lambda }_{1}-2m\hspace{0.1em}\text{/}\hspace{0.1em}n, where λ 1 {\lambda }_{1} is the largest eigenvalue of the adjacency matrix of G G , and 2 m / n 2m\hspace{0.1em}\text{/}\hspace{0.1em}n is the average vertex degree of G G . Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer i ≥ 0 i\ge 0 , it is shown that there exists a graph G G such that CS ( G ) = i \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)=i . Also, for every interval I i = ( i , i + 1 ) {I}_{i}=\left(i,i+1) , it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in I i {I}_{i} .

The sigma-irregularity index is a natural variant of the well-established Albertson irregularity index. Here, we introduce an irregularity measure based on the sigma-irregularity, which is a graph invariant with respect to a given degree sequence. We define it as sigma(t) (G ) = 1/2 Sigma(v,wV(G))(d(G)(v) - d(G)(w))(2), where d(G)(v) is the degree of a vertex v of G , and named it the total sigma-irregularity. We characterize irregular graphs with minimal sigma(t)-irregularity. In addition, we consider the so-called inverse problem for the Albertson irregularity index, the total irregularity, and the sigma(t)-irregularity. For those irregularity measures, we study the problem for general graphs, trees, and c-cyclic graphs. (c) 2022 Elsevier Inc. All rights reserved.

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set {x1,…,xn}, then the ad-hoc Lanzhou index of G is defined by Lz˜(G)=∑i=1ndi(n−1−di)2, where di represents the degree of the vertex xi. Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ-cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ-cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.

Fora graph G of order n, size m and degree sequence D (G) = (d1,d2,...,dn), a new measure of irregularity IAG(G) = 1 - nn(d1 + r)(d2+ r) center dot center dot center dot (dn + r)/(2m + rn)n, r is an element of R?,0, is introduced. It is shown that if G has maximum IAG-irregularity among all con-nected graphs of order n and size m, then (i) Delta(G) = n - 1; (ii) for each u,v is an element of V(G) with the property dG(u) 5 dG(v), it holds that N(G,u) subset of N[G,v], where N(G,w) and N[G,w] are the neighbourhood and the closed neighbourhood of w in G, respectively; (iii) G is a threshold graph. Further, it is proven that if a graph H has a minimum value of IAG-irregularity among all irregular graphs of the same order and size, then Delta(H) - delta (H) = 1. Finally, the graphs with minimum and maximum IAG-irregularity in the classes of trees, unicyclic and bicyclic graphs are characterized.

We provide new families of divisibility and strong divisibility sequences based on some factorization properties of Chebyshev polynomials.

The atom–bond connectivity (ABC) index was introduced in the last quarter of the 1990s to improve the prediction power of the Randić index. Later on, in 2008, the factor √ 2 was dropped from the original definition of the ABC index, and some additional chemical applications of this index were reported, which resulted in considerable interest in studying the mathematical properties of the ABC index. There are more than a hundred papers devoted to the mathematical aspects of this graph invariant. The primary purpose of this review is to gather the existing bounds and extremal results concerning the ABC index.