Tricyclic Graphs Research Articles

The Singularity of Three Kinds of New Tricyclic Graphs

A graph G is singular if its adjacency matrix is singular. The starting vertices of two paths Pb1 and Pb2 are simultaneously bound to the ending vertex of the path Ps1, and the ending vertices of the paths Pb1 and Pb2 are bound to the starting vertex of path Ps2. Meanwhile, the starting vertex of the path Ps1 is bound to a vertex of the cycle Ca1, and the ending vertex of the path Ps2 is bound to a vertex of the cycle Ca2. Thus, the resulting graph is written as ξ(a1,a2,b1,b2,s1,s2). This is denoted by ζ(a1,a2,b1,b2,s)=ξ(a1,a2,b1,b2,1,s) and ε(a1,a2,b1,b2)=ζ(a1,a2,b1,b2,1), which are referred to as the ξ-graph, ζ-graph and ε-graph for short, respectively. It is known that there are 15 kinds of tricyclic graphs. The purpose of this paper is to study the necessary and sufficient conditions for ξ-graphs, ζ-graphs and ε-graphs to be singular graphs. We analyzed the structure of the elementary spanning subgraphs of the graph G=ξ(a1,a2,b1,b2,s1,s2). By calculating the determinant of the adjacency matrix of the graph G, the necessary and sufficient conditions for the determinant of the graph G to be zero is obtained, and so the necessary and sufficient conditions for graph ξ(a1,a2,b1,b2,s1,s2) to be singular are obtained. As the corollaries, the necessary and sufficient conditions for graphs ζ(a1,a2,b1,b2,s) and ε(a1,a2,b1,b2) to be singular are also obtained.

On Sombor index and graph energy of some chemically important graphs

Sombor index of a graph G=(V(G),E(G)) is provided by the expression ∑uv∈E(G)du2+dv2, where dx is the degree of the vertex x∈V(G). The energy of a graph is the quantity given by the total of the absolute values of its adjacency matrix’s eigenvalues. In this article, we improve the relation between the Sombor index and graph energy and derive the relation between them for unicyclic, bicyclic and tricyclic graphs, trees, triangular chain, square cactus chain and hexagonal cactus chain graphs. At last, we find the bounds of graph energy for zigzag and linear hexagonal chains.

On tricyclic graphs with maximum atom–bond sum–connectivity index

The sum-connectivity, Randić, and atom-bond connectivity indices have a prominent place among those topological indices that depend on the graph's vertex degrees. The ABS (atom-bond sum-connectivity) index is a variant of all the aforementioned three indices, which was recently put forward. Let T(n) be the class of all connected tricyclic graphs of order n. Recently, the problem of determining graphs from T(n) having the least possible value of the ABS index was solved in (Zuo et al., 2024 [39]) for the case when the maximum degree of the considered graphs does not exceed 4. The present paper addresses the problem of finding graphs from T(n) having the largest possible value of the ABS index for n≥5.

On unimodular graphs with a unique perfect matching

A graph is called unimodular if its adjacency matrix has determinant ±1. This article provides a necessary and sufficient condition for a simple connected graph with a unique perfect matching to be unimodular. In particular, we give a complete characterization of bicyclic unimodular graphs with a unique perfect matching. Moreover, the possible values of the determinant of the adjacency matrix of unicyclic, bicyclic, and tricyclic graphs with a unique perfect matching are also provided in this article. For non-bipartite unicyclic graphs with a unique perfect matching, we address the problem of when the inverse of the corresponding adjacency matrix is diagonally similar to a non-negative matrix. A pseudo-unimodular graph is a singular graph whose product of non-zero eigenvalues of the corresponding adjacency matrix is ±1. We supply a necessary and sufficient condition for a singular graph to be pseudo-unimodular.

Tricyclic Graph with Minimum Randić Index

The Randić index of a graph G is the sum of (dG(u)dG(v))−12 over all edges uv of G, where dG(u) denotes the degree of vertex u in G. In this paper, we investigate a few graph transformations that decrease the Randić index of a graph. By applying those transformations, we determine the minimum Randić index on tricyclic graphs and characterize the corresponding extremal graphs.

On the atom-bond sum-connectivity index of chemical graphs

The atom-bond sum-connectivity (ABS) index of a given graph G is symbolized with ABS and define by sum of the weights du+dv−2du+dv, where du is the degree of a vertex u in G. In this paper, the extremal atom-bond sum-connectivity index among all n-vertex chemical trees, connected chemical unicyclic graphs, connected chemical bicyclic graphs, and connected chemical tricyclic graphs, were determined.

Counting independent sets in tricyclic graphs

Counting independent sets in tricyclic graphs

Conjugated tricyclic graphs with maximum variable sum exdeg index

The variable sum exdeg index, initially introduced by Vukicevic (2011) [20] for predicting the octanol water partition co-efficient of certain chemical compounds, is an invariant for a graph G and defined as SEIa(G)=∑v∈V(G)(dvadv), where dv is the degree of vertex v∈V(G), a is a positive real number different from 1. In this paper, we defined sub-collections of tricyclic graphs say T2m3,T2m4,T2m6 and T2m7. The graph with maximum variable sum exdeg index is characterized from each collection given above with perfect matching. Consequently, through a comparison among these extremal graphs, we indicate the graph which contains maximum SEIa-value from T2m.

The weighted vertex PI index of tricyclic graphs

The weighted vertex PI index of tricyclic graphs

Hitting times for random walks on tricyclic graphs

Let G be a simple connected graph and Let be the expected hitting time from x to y in G and be the hitting time of G, where A tricyclic graph is a simple connected graph that the edge number equals the vertex number plus two. Let be the set of all n-vertex tricyclic graphs. In this article, we will determine the extremal graphs for hitting times among all n-vertex tricyclic graphs. Moreover, if then we will obtain sharp upper and lower bounds for

On Sombor Indices of Tricyclic Graphs

This paper indicates alternative ways of deriving the main results of the recent paper [M. Zhang, B. Zhao, Extremal values of the Sombor index in tricyclic graphs, MATCH Commun. Math. Comput. Chem. 89 (2023) 741-758]. The graphs possessing minimum and maximum values of the reduced Sombor index are also characterized from the class of all tricyclic connected graphs of a given order.

Extremal Values of the Sombor Index in Tricyclic Graphs

Extremal Values of the Sombor Index in Tricyclic Graphs

The singularity of two kinds of tricyclic graphs

<abstract><p>Let $ G $ be a finite simple graph and let $ A(G) $ be its adjacency matrix. Then $ G $ is $ singular $ if $ A(G) $ is singular. Suppose $ P_{b_{1}}, P_{b_{2}}, P_{b_{3}} $ are three paths with disjoint vertices, where $ b_i\geq 2 (i = 1, 2, 3) $, and at most one of them is 2. Coalescing together one of the two end vertices of each of the three paths, and coalescing together the other end vertex of each of the three paths, the resulting graph is called the $ \theta $-graph, denoted by $ \theta(b_{1}, b_{2}, b_{3}) $. Let $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ be the graph obtained by merging one end of the path $ P_{s} $ with one vertex of a cycle $ C_{a} $, and merging the other end of the path $ P_{s} $ with one vertex of $ \theta(b_{1}, b_{2}, b_{3}) $ of degree 3. If $ s = 1 $, denote $ \beta(a, b_{1}, b_{2}, b_{3}) = \alpha(a, b_{1}, b_{2}, b_{3}, 1) $. In this paper, we give the necessity and sufficiency condition for the singularity of $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ and $ \beta(a, b_{1}, b_{2}, b_{3}) $, and we also prove that the probability that any given $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ is a singular graph is equal to $ \frac{35}{64} $, the probability that any given $ \beta(a, b_{1}, b_{2}, b_{3}) $ is a singular graph is equal to $ \frac{9}{16} $. From our main results we can conclude that such a $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is singular if $ 4|a $ or three $ b_i (i = 1, 2, 3) $ are all odd numbers or exactly two of the three $ b_i (i = 1, 2, 3) $ are odd numbers and the length of the cycle formed by the two odd paths in $ \alpha(a, b_{1}, b_{2}, b_{3}, s) $ graph ($ \beta(a, b_{1}, b_{2}, b_{3}) $ graph) is a multiple of 4. The theoretical probability of these graphs being singular is more than half.</p></abstract>

On the Merrifield–Simmons index of tricyclic graphs

The total number of independent subsets, including the empty set, of a graph, is also termed as the Merrifield–Simmons index (MSI) in mathematical chemistry. Zhu and Yu (2012) presented a lower bound on Merrifield–Simmons index of tricyclic graphs in terms of order n and the characterization of extremal graphs. This result was erroneous. In this paper, we correct this result. Moreover, we characterize the tricyclic graphs on n vertices with the second and third smallest values of the Merrifield–Simmons index.

The Singularity of Four Kinds of Tricyclic Graphs

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths Ps1, Ps2, Ps3 into a vertex, and three end vertices into a vertex on the cycle Ca1, Ca2, Ca3, respectively, is denoted as γ(a1,a2,a3,s1,s2,s3). Note that δ(a1,a2,a3,s1,s2)=γ(a1,a2,a3,s1,1,s2), ζ(a1,a2,a3,s)=γ(a1,a2,a3,1,1,s), φ(a1,a2,a3)=γ(a1,a2,a3,1,1,1). In this paper, we give the necessity and sufficiency that the γ−graph, δ−graph, ζ−graph and φ−graph are singular and prove that the probability that a randomly given γ−graph, δ−graph, ζ−graph or φ−graph being singular is equal to 325512,165256,4364, 2132, respectively. From our main results, we can conclude that such a γ−graph(δ−graph, ζ−graph, φ−graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems.

Minimum Total Irregularity Index Of The Tricyclic Graphs

The quantitative characterization of the topological structures of irregular graphs has been demonstrated through several irregularity measures. In the literature, not only different chemical and physical properties can be well comprehended but also quantitative structure-activity relationship (QSPR) and quantitative structure-property relationship (QSAR) are documented through these measures. A simple graph G = (V;E) is a collection of V and E as vertex and edge sets respectively, with no multiple edges or loops. Keeping in view the importance of various irregularity measures, in (Abdo and Dimitrov 2012) the authors defined the total irregularity of a simple graph G = G(V;E) as irrt(G) = 1 2 P u;v2V jdG(u) 􀀀 dG(v)j; where dG(u) indicates the degree of the vertex u, where u 2 V (G). In this paper, we have determined the first minimum, second minimum and third minimum total irregularity index of the tricyclic graphs on the n vertices.

Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

On Omega Index and Average Degree of Graphs

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called o m e g a index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to k -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.

The A α -spectral radius of complements of bicyclic and tricyclic graphs with n vertices

Abstract Recently, the extremal problem of the spectral radius in the class of complements of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs had been studied widely. In this paper, we extend the largest ordering of A α -spectral radius among all complements of bicyclic and tricyclic graphs with n vertices, respectively.

Maximum total irregularity index of some families of graph with maximum degree n − 1

The total irregularity index of a graph [Formula: see text] is defined by Abdo et al. [H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014) 201–206] as [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In 2014, You et al. [L. H. You, J. S. Yang and Z. F. You, The maximal total irregularity of unicyclic graphs, Ars Comb. 114 (2014) 153–160.] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Unicyclic graphs) and Zhou et al. [L. H. You, J. S. Yang, Y. X. Zhu and Z. F. You, The maximal total irregularity of bicyclic graphs, J. Appl. Math. 2014 (2014) 785084, http://dx.doi.org/10.1155/2014/785084] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Bicyclic graphs). In this paper, we characterize the aforementioned graphs with an alternative but comparatively simple approach. Also, we characterized the graphs having maximum [Formula: see text] value among the classes [Formula: see text] (Tricyclic graphs), [Formula: see text] (Tetracyclic graphs), [Formula: see text] (Pentacyclic graphs) and [Formula: see text] (Hexacyclic graphs).