Generalized Version of ISI Invariant for some Molecular Structures

Degree related topological invariants are the bygone and most victorioustype of graph invariants so far. In this article, we are interested in finding the generalized inverse indeg invariant of the nanostar dendrimers D[r],fullerene dendrimerNS4[r], and polymer dendrimerNS5[r]. Keywords: nanotubes; inverse indeg invariant; nanostar dendrimers; fullerene dendrimer; polymer dendrimer

Estimating vertex-degree-based energies

Introduction/purpose: In the current literature, several dozens of vertex-degree-based (VDB) graph invariants are being studied. To each such invariant, a matrix can be associated. The VDB energy is the energy (= sum of the absolute values of the eigenvalues) of the respective VDB matrix. The paper examines some general properties of the VDB energy of bipartite graphs. Results: Estimates (lower and upper bounds) are established for the VDB energy of bipartite graphs in which there are no cycles of size divisible by 4, in terms of ordinary graph energy. Conclusion: The results of the paper contribute to the spectral theory of VDB matrices, especially to the general theory of VDB energy.

Fault-Tolerant Partition Resolvability of Cyclic Networks

Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.

On edge-degree based invariants of graphs

Let [Formula: see text] be a graph with edge set [Formula: see text]. For an edge [Formula: see text] in [Formula: see text], we define [Formula: see text], where [Formula: see text] and [Formula: see text] are degrees of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. For [Formula: see text], the graph invariants [Formula: see text], [Formula: see text] and [Formula: see text] are defined as [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are incident. In this paper, some relationship between these graph invariants and some classical topological indices were presented. Moreover, some bounds for [Formula: see text], [Formula: see text] and [Formula: see text] are obtained and trees with the first through the third smallest [Formula: see text] and [Formula: see text], as well as the trees with the first through the forth smallest [Formula: see text] are also characterized.

Fast Diameter Computation within Split Graphs

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.

Some properties of subgroup complementary addition Cayley graphs on abelian groups

In this paper, we introduce subgroup complementary addition Cayley graph [Formula: see text] and compute its graph invariants. Also, we prove that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text] where [Formula: see text].

Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G , its vertex-degree-based topological indices of the form BID G = ∑ u v ∈ E G β d u , d v are known as bond incident degree indices, where E G is the edge set of G , d w denotes degree of an arbitrary vertex w of G , and β is a real-valued-symmetric function. Those BID indices for which β can be rewritten as a function of d u + d v − 2 (that is degree of the edge u v ) are known as edge-degree-based BID indices. A connected graph G is said to be r -apex tree if r is the smallest nonnegative integer for which there is a subset R of V G such that R = r and G − R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r -apex trees of order n , where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1 .

On a Conjecture about the Sombor Index of Graphs

Let G be a graph with vertex set V(G) and edge set E(G). A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 and SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) is the degree of the vertex v in G. We denote by Hn,ν the graph constructed from the star Sn by adding ν edge(s), 0≤ν≤n−2, between a fixed pendent vertex and ν other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph Hn,ν has the maximum Sombor index among all connected ν-cyclic graphs of order n, where 0≤ν≤n−2. In some earlier works, the validity of this conjecture was proved for ν≤5. In this paper, we confirm that this conjecture is true, when ν=6. The Sombor index in the case that the number of pendent vertices is less than or equal to n−ν−2 is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained.

On Average Distance of Neighborhood Graphs and Its Applications

Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then and μG1′≤μG and μG2′≤μG whenever G1′ and G2′ are components of G′. In order to make our claims substantial and determine graphs for which the bounds are best possible, we performed some experiments in MATLAB software. Simulation results agree very well with the propositions and proofs. Finally, we have described how our results may be applied in socio-epidemiology and ecology and then concluded with other proposed further research questions.

Joining Forces for Reconstruction Inverse Problems

A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.