‎The Laplacian eigenvalues and polynomials of the networks play an essential role in understanding the relations between the topology and the dynamic of networks‎. ‎Generally‎, ‎computation of the Laplacian spectrum of a network is a hard problem and there are just a few classes of graphs with the property that their spectra have been completely computed‎. ‎Laplacian spectrum for $ n$-prism networks was investigated in [Liu et al.‎, ‎Neurocomputing 198 (2016) 69-73]‎. ‎In this paper‎, ‎we give a method for calculating the eigenvalues and characteristic polynomial of the Laplacian matrix of a generalized $n$-prism network‎. ‎We show how such large networks can be constructed from small graphs by using graph products‎. ‎Moreover‎, ‎our results are used to obtain the Kirchhoff index and the number of the spanning trees in the generalized $n$-prism networks‎. ‎We also give some examples of applications‎, ‎that explain the usefulness and efficiency of the proposed method‎.

‎Bucket recursive trees are an interesting and natural generalization of recursive trees‎. ‎In this model the nodes are buckets that can hold up to b≥ 1 labels‎. ‎The (modified) Zagreb index of a graph is defined as the sum of‎ ‎the squares of the outdegrees of all vertices in the graph‎. ‎We give the mean and variance of this index in random bucket recursive trees‎. ‎Also‎, ‎two limiting results on this index are given‎.

‎Since the problem of computing the adjacency dimension of a graph is NP-hard‎, ‎finding the adjacency dimension of special classes of graphs or obtaining good bounds on this invariant is valuable‎. ‎In this paper we determine the properties of each adjacency resolving set of paths. ‎Then, ‎by ‎using ‎these ‎properties,‎ we determine the adjacency dimension of broom and double broom graphs‎.

‎For a graph G‎, ‎the exponential reduced Sombor index (ERSI)‎, ‎denoted by eSored , ‎is ∑‎uv∈E(G) e√(dG(v)-1)^2+(dG(u)-1)^2), ‎where dG(v) is the degree of vertex v‎. ‎The authors in [On the reduced Sombor index and its applications‎, ‎MATCH Commun‎. ‎Math‎. ‎Comput‎. ‎Chem‎. ‎86 (2021) 729–753] conjectured that for each molecular tree T of order n‎,  eSored‎(T)≤(2/3) (n+1) e3 +(1/3) (n-5) e 3√2, where n≡2 (mod 3), eSored‎(T)≤(1/3) (2n+1) e3 +(1/3) (n-13) e3√2 + 3e√13 , where n≡1 (mod 3) and  eSored‎(T)≤(2/3) ne3 +(1/3) (n-9) e3√2 + 2e√10 , where n≡0 (mod 3). ‎Recently‎, ‎Hamza and Ali [On a conjecture regarding the exponential reduced Sombor index of chemical trees‎. ‎Discrete Math‎. ‎Lett‎. ‎9 (2022) 107–110] proved the modified version of this conjecture‎. ‎In this paper‎, ‎we adopt another method to prove it‎.

‎Zagreb indices were reformulated in terms of the edge degrees instead of the vertex degrees‎. For a graph $G$‎, ‎the first and second reformulated Zagreb indices are defined respectively as‎:‎$$EM_1(G)=\sum_{\varepsilon\in E(G)}d^2(\varepsilon), EM_2(G)=\sum_{\varepsilon,\varepsilon'\in E(G),\,\varepsilon\sim \varepsilon'}d(\varepsilon)\,d(\varepsilon'),$$‎ where $d(\varepsilon)$ and $d(\varepsilon')$ denote the degree of the edges $\varepsilon$ and $\varepsilon'$ respectively‎, ‎and $\varepsilon\sim \varepsilon'$ means that the edges $\varepsilon$ and $\varepsilon'$ are adjacent‎. In this paper‎, ‎we obtain sharp lower bounds on the first and second reformulated Zagreb‎ indices with a given number of vertices and maximum degree‎. ‎Furthermore‎, ‎we will determine the extremal trees that achieve these lower bounds‎.

‎New definitions employing the golden ratio as the characteristic parameter are proposed. The definitions are classified into two categories: Geometrical and Physical properties‎. ‎In the first category‎, ‎the golden ratio tree is defined‎, ‎and its properties are discussed through theorems‎. ‎Then‎, ‎decaying and growing type golden ratio spirals are proposed and discussed‎. ‎The equation producing the golden ratio heart in the analytical two-dimensional space is given‎. ‎Regarding the second category‎, ‎the golden ratio ball is defined with respect to collisions with the ground and the collision coefficient is determined‎. ‎Golden ratio damping is another new definition in which the dimensionless damped parameter is determined in terms of the golden ratio‎. ‎Theorems are posed and proven regarding the properties of the definitions‎. ‎Numerical solutions in the form of plots are given when necessary‎.

The main goal of this paper is to study the modified $F$-indices (modified first Zagreb index and modified forgotten topological index) of random $m$-oriented recursive trees (RMORTs). First, through two recurrence equations, we compute the mean and the variance of these indices in our random tree model. Second, we show four convergence in probability based on these indices. Third, the asymptotic normalities, through the martingale central limit theorem, are given.

A hypertree is a special type of connected hypergraph that removes‎ ‎any‎, ‎its hyperedge then results in a disconnected hypergraph‎. ‎Relation between hypertrees (hypergraphs) and trees (graphs) can be helpful to solve real problems in hypernetworks and networks and it is the main tool in this regard‎. ‎The purpose of this paper is to introduce a positive relation (as $\alpha$-relation) on hypertrees that makes a connection between hypertrees and trees‎. ‎This relation is dependent on some parameters such as path‎, ‎length of a path‎, ‎and the intersection of hyperedges‎. ‎For any $q\in \mathbb{N}‎, ‎$ we introduce the concepts of a derivable Tree‎, ‎$(\alpha‎, ‎q)$-hypergraph‎, ‎and fundamental $(\alpha‎, ‎q)$-hypertree for the first time in this study and analyze the structures of derivable trees from hypertrees via given positive relation‎. ‎In the final‎, ‎we apply the notions of derivable trees‎, ‎$(\alpha‎, ‎q)$-trees in real optimization problems by modeling hypernetworks and networks based on hypertrees and trees‎, ‎respectively.‎‎‎

Graphs are an effective tool for planetary gear trains (PGTs) synthesis and for the enumeration of all possible PGTs for transmission systems. In the past fifty years, considerable effort has been devoted to the synthesis of PGTs. To date, however, synthesis results are inconsistent, and accurate synthesis results are difficult to achieve. This paper proposes a systematic approach for synthesizing PGTs depending on spanning trees and parent graphs. Trees suitable for constructing rooted graphs are first identified. The parent graphs are then listed. Finally, geared graphs are discovered by inspecting their parent graphs and spanning trees. To precisely detect spanning trees, a novel method based on two link assortment equations is presented. Transfer vertices and edge levels are detected without the use of any computations. This work develops the vertex matrix of the rooted graph, and its distinctive equation is used to arrange the vertex degree arrays according to the vertex levels and eliminate the arrays that violate the distinctive equations. The precise results of the 5-link geared graphs are confirmed to be 24. The disparity between the recent and previous synthesis results can be attributed to the fact that the findings of the current method, which employs rooted graphs, are more comprehensive than those obtained with graphs lacking multiple joints. A novel algorithm for detecting structural isomorphism is proposed. By comparing the vertex degree listings and gear strings, non-isomorphic geared graphs are obtained. The algorithm is simple and computationally efficient. The graph representation is one-to-one with the vertex degree listing and gear string representation. This allows for the storage of a large number of graphs on a computer for later use.

‎For a simple graph G‎, ‎the Gordon-Scantlebury index of G is equal to the number of paths of length two in G‎, ‎and the Platt index is equal to the total sum of the degrees of all edges in G‎. ‎In this paper‎, ‎we study these indices in random plane-oriented recursive trees through a recurrence equation for the first Zagreb index‎. ‎As n ∊ ∞, ‎the asymptotic normality of these indices are given‎.