The metallic and the Horadam cubes are two recently introduced generalizations of Fibonacci cubes. They preserve many properties that make the Fibonacci cubes relevant for various applications. In this paper, we show that both the metallic and the Horadam cubes appear as the resonance graphs of well-known and chemically realistic families of benzenoid and phenylene compounds. Along the way, we also address the realizability problem for Horadam sequences regarding the number of perfect matchings. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we confirm a conjecture by Furtula and Oz regarding graphs that maximize the second complementary Zagreb index. We demonstrate that this conjecture holds for a broader class of indices, each of which is parameter-dependent, and which we will refer to as the generalized complementary second Zagreb index. It is shown that all indices in this class are maximized by complete split graphs. Additionally, we analyze the behavior of the clique order in optimal graphs. For the case of the second complementary Zagreb index, we provide an explicit expression, thereby confirming the value conjectured by Furtula and Oz.

Catastrophic fault patterns are failure patterns resulting in the loss of connectivity of a processor array. We provide bijective correspondences between such patterns in one-dimensional processor arrays and lattice paths satisfying certain inertial criteria. In that way, we obtain new results for the asymptotic behavior of the enumerating sequences of catastrophic fault patterns.We also recover and reinterpret several known enumerative results and open possibilities for obtaining analogous results for more complex, but still essentially one-dimensional arrays.

The Kirchhoff index Kf(G) of a connected graph G is defined as the sum of resistande distances between all pairs of vertices in G. We say that v is an element of V (G) is a good vertex if the Kirchhdff index re-mains unchanged when is removed, Le. Kf(G)=Kf(G-v). In 1991, Soltes studied the Wiener index of a graph and posed the problem of identifying graphs for which the removal of an arbitrary vertex preserves the Wiener index. In this paper, we explore a similar concept: identifying Kirch-hoffSoltes graphs, i.e. graphs in which all vertices are good vertices. We show that the cycle C-5, is a KirchhoffSoltes graph. Due to the challenge of finding more examples of such graphs, we shift our focus to several relaxed versions of the KirchhoffSoltes problem, where the primary objec-tive is to identify graphs containing at least one good vertex. One of them is the beta-KirchhoffSoltes problem, which seeks to find an infinite family of graphs in which the proportion of good vertices is at least beta, with is an element of (0, 1] being a specified rational number. Another one involves constructing infinite families of graphs where the proportion of good vertices increases and asymptotically approaches a given real number y is an element of (0, 1) as the order of the graph grows. We demonstrate that both relaxed versions have infinitely many solutions. In particular, we prove the existence of infinitely many graphs for which the proportion of good vertices, 1/7 <=beta < 1/5 tends to a cer-tain irrational number. Furthermore, we prove the existence of infinitely many graphs with half good vertices, and for cach s is an element of N, we construct an infinite family of graphs whose proportion of good vertices tends to (S+ 1)/(2S + 1) These findings could be pivotal in addressing the original problem of determining whether are additional solutions beyond C-5.

In this note we collect and state some facts on the parity of some integer-valued graph invariants. In particular, we show that several well-known bond-additive topological indices assume only even values for all connected graphs. Among them are both degree-based and distance-based indices. We also discuss some general graph-theoretical consequences of established results.

The Riviera model is a combinatorial model for a settlement along a coastline, introduced recently by the authors. Of most interest are the so-called jammed states, where no more houses can be built without violating the condition that every house needs to have free space to at least one of its sides. In this paper, we introduce new agents (predators and altruists) that want to build houses once the settlement is already in the jammed state. Their behavior is governed by a different set of rules, and this allows them to build new houses even though the settlement is jammed. Our main focus is to detect jammed configurations that are resistant to predators, to altruists, and to both predators and altruists. We provide bivariate generating functions, and complexity functions (configurational entropies) for such jammed configurations. We also discuss this problem in the two-dimensional setting of a combinatorial settlement planning model that was also recently introduced by the authors, and of which the Riviera model is just a special case.

This paper considers the generalized atom-bond sum-connectivity index 𝒜ℬ𝒮_ for 1≤ < 2 and gives the characterization of those graphs that minimize this index on the class of all fixed-order unicyclic graphs of a given maximum degree Δ. It is also proved that the cycle graph Cn uniquely minimizes the aforementioned index in the class of all fixed-order unicyclic graphs. The obtained results imply conclusions concerning the maximum values of the well-known harmonic index over the considered class of graphs. When the class of investigated graphs is limited to the class of molecular unicyclic graphs, all the obtained results are still valid.

For a given graph G with n vertices and m edges, let M2 denote the second Zagreb index of G and lambda 1 its spectral radius. We show that the inequality M2m <= 2 lambda 21 is not valid in general case, but is satisfied by trees and unicyclic graphs, settling thus an old conjecture by one of the present authors. We also pose some related problems for future research.

The Harary index and the Wiener index are two well-studied topological indices in chemical graph theory. Quite recently, the graphs attaining the minimum value of the product of the Harary and Wiener indices were characterized in [E. Azjargal, B. Horoldagva, I. Gutman, Minimum of product of Wiener and Harary indices, MATCH Commun. Math. Comput. Chem. 92 (2024) 65-71] over the class of all connected graphs of a fixed order and size. The present paper provides a generalization, involving Wiener-type topological indices and their reciprocals, of the aforementioned result.

This article is concerned with qualitative and quantitative refinements of the concepts of the log-convexity and log-concavity of positive sequences. A new class of tempered sequences is introduced, its basic properties are established and several interesting examples are provided. The new class extends the class of log-balanced sequences by including the sequences of similar growth rates, but of the opposite log-behavior. Special attention is paid to the sequences defined by two- and three-term linear recurrences with constant coefficients. For the special cases of generalized Fibonacci and Lucas sequences, we graphically illustrate the domains of their log-convexity and log-concavity. For an application, we establish the concyclicity of the points a2na2n+1,1a2n+1 for some classes of Horadam sequences (an) with positive terms.

A nonnegative integer $p$ is realizable by a graph-theoretical invariant $I$ if there exist a graph $G$ such that $I(G) = p$. The inverse problem for $I$ consists of finding all nonnegative integers $p$ realizable by $I$. In this paper, we consider and solve the inverse problem for the Mostar index, a recently introduced graph-theoretical invariant which attracted a lot of attention in recent years in both the mathematical and the chemical community. We show that a nonnegative integer is realizable by the Mostar index if and only if it is not equal to one. Besides presenting the complete solution to the problem, we also present some empirical observations and outline several open problems and possible directions for further research.

The ABS (atom-bond sum-connectivity) index is a topological index, that was introduced in 2022 by amalgamating the main ideas of two well-examined indices. Mathematical aspects (especially, extremal results and bounds) of the ABS index have already been studied considerably. The primary goal of this review paper is to collect known bounds and extremal results regarding the ABS index. Several new extremal results, which follow easily from existing general results, are also given. Moreover, a number of open problems and conjectures, arising from the reported results, are proposed.

A matching in a graph G is a collection of edges of G such that no two of them share a vertex. The number of all matchings in G is called its Hosoya index. In this paper, we compute Hosoya indices of several classes of unbranched polymers made of cycles of the same lengths arranged around a middle path and decorated by attaching to each vertex, a given number of pendent vertices or thorns. We establish linear recurrences satisfied by those numbers and obtain explicit formulas in terms of Fibonacci polynomials and their generalizations. Some possible directions of future research are also indicated.

In this article, we determine the complexity function (configurational entropy) of jammed configurations of Rydberg atoms on a one-dimensional lattice. Our method consists of providing asymptotics for the number of jammed configurations determined by direct combinatorial reasoning. In this way we reduce the computation of complexity to solving a constrained optimization problem for the Shannon's entropy function. We show that the complexity can be expressed explicitly in terms of the root of a certain polynomial of degree $b$, where $b$ is the so-called blockade range of a Rydberg atom. Our results are put in a relation with the model of irreversible deposition of $k$-mers on a one-dimensional lattice.

A matching M in a graph G is an induced matching if the largest degree of the subgraph of G induced by M is equal to one. A dominating induced matching (DIM) of G is an induced matching that dominates every edge of G. It is well known that, if they exist, all dominating induced matchings of G are of the same size. The dominating induced matching number of G, denoted by dim(G), is the size of any dominating induced matching of G. In this paper, we continue the study of dominating induced matchings. We prove that, if G has a DIM, then the induced matching number of G is equal to the independence number of its line graph L(G) and to the edge domination number of G. It is also shown that dim(G) <= 2 dim(L(G)), provided that both G and L(G) have a DIM. We also present some bounds on dim(G). In particular, for a tree T with a DIM we show that (sic)n-l+1/3(sic) <= dim(T) <= (sic)n-1+l/3(sic), where l is the number of leaves. Moreover, for a regular graph G we establish some Nordhaus-Gaddum type bounds.

In random sequential adsorption (RSA), objects are deposited on a substraterandomly, irreversibly, and sequentially. Attempts of deposition that lead to anoverlap with previously deposited objects are discarded. The process continuesuntil the system reaches a jammed state when no further additions are possible.We analyze a class of RSA models on a two-row square ladder graph in whichlanding on an empty site in a graph is allowed when at leastbneighboring sitesin the graph are unoccupied (b is an element of N). In this paper we complement this typicalway of studying RSA models by analyzing also the structure of the set of alljammed states in a static way, disregarding the dynamics that led to a particularjammed state. In both considered settings (dynamic and static) we provideexplicit expressions for key statistics that describe the average proportion ofthe substrate covered by deposited objects, and then we comment on significantdifferences between the two settings. We illustrate all of our findings througha toy model for ensembles of trapped Rydberg atoms with blockade range b.

The number of spanning trees of a graph G is called the complexity of G. It is known that the complexity of the line graph of a given graph G can be computed as the sum over all spanning trees of G of contributions which depend on various types of products of degrees of vertices of G. We interpret the contributions in terms of three types of multiplicative Zagreb indices, obtaining simple and compact expressions for the complexity of line graphs of graphs with low cyclomatic numbers. As an application, we determine the unicyclic graphs whose line graphs have the smallest and the largest complexity. (C) 2024 University of Kashan Press. All rights reserved.