scholarly journals On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2395
Author(s):  
Wenjie Ning ◽  
Kun Wang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Estrada Index

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

scholarly journals Unicyclic graphs with given number of cut vertices and the maximal Merrifield-Simmons index

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 451-461 ◽  
Author(s):  
Hongbo Hua ◽  
Xinli Xu ◽  
Hongzhuan Wang
Keyword(s):  
Connected Graph ◽  
Independent Sets ◽  
Unicyclic Graph ◽  
Unicyclic Graphs

The Merrifield-Simmons index of a graph G, denoted by i(G), is defined to be the total number of independent sets in G, including the empty set. A connected graph is called a unicyclic graph, if it possesses equal number of vertices and edges. In this paper, we characterize the maximal unicyclic graph w.r.t. i(G) within all unicyclic graphs with given order and number of cut vertices. As a consequence, we determine the connected graph with at least one cycle, given number of cut vertices and the maximal Merrifield-Simmons index.

Tetracyclic graphs with maximal Estrada index

2017 ◽  
Vol 09 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Nader Jafari Rad ◽  
Akbar Jahanbani ◽  
Doost Ali Mojdeh
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Estrada Index ◽  
Simple Connected Graph

The Estrada index of a simple connected graph [Formula: see text] of order [Formula: see text] is defined as [Formula: see text], where [Formula: see text] are the eigenvalues of the adjacency matrix of [Formula: see text]. In this paper, we characterize all tetracyclic graphs of order [Formula: see text] with maximal Estrada index.

scholarly journals Binomial edge ideals of unicyclic graphs

2021 ◽  
pp. 1-26
Author(s):  
Rajib Sarkar
Keyword(s):  
Betti Number ◽  
Connected Graph ◽  
Betti Numbers ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Edge Ideals ◽  
Vertex Set ◽  
Bounded Below ◽  
Extremal Betti Number

Let [Formula: see text] be a connected graph on the vertex set [Formula: see text]. Then [Formula: see text]. In this paper, we prove that if [Formula: see text] is a unicyclic graph, then the depth of [Formula: see text] is bounded below by [Formula: see text]. Also, we characterize [Formula: see text] with [Formula: see text] and [Formula: see text]. We then compute one of the distinguished extremal Betti numbers of [Formula: see text]. If [Formula: see text] is obtained by attaching whiskers at some vertices of the cycle of length [Formula: see text], then we show that [Formula: see text]. Furthermore, we characterize [Formula: see text] with [Formula: see text], [Formula: see text] and [Formula: see text]. In each of these cases, we classify the uniqueness of the extremal Betti number of these graphs.

scholarly journals Further Results on the Resistance-Harary Index of Unicyclic Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 201 ◽  
Author(s):  
Jian Lu ◽  
Shu-Bo Chen ◽  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Ying-Jie Ji
Keyword(s):  
Connected Graph ◽  
Unicyclic Graph ◽  
Resistance Distance ◽  
Harary Index ◽  
Unicyclic Graphs

The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.

On the Estrada index of cacti

2014 ◽  
Vol 27 ◽  
Author(s):  
Erfang Shan ◽  
Hongzhuan Wang ◽  
Liying Kang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Common Vertex ◽  
Estrada Index ◽  
Unique Graph ◽  
Simple Connected Graph

Let G be a simple connected graph on n vertices and λ_1, λ_2, . . . , λ_n be the eigenvalues of the adjacency matrix of G. The Estrada index of G is defined as EE(G) = \sum_{i=1}^n e^{λi}. A cactus is a connected graph in which any two cycles have at most one common vertex. In this work, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cycles was determined. Also, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cut edges was determined.

Extermal forgotten topological index of quasi-unicyclic graphs

2021 ◽  
pp. 2250041
Author(s):  
Amir Taghi Karimi
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Upper And Lower Bounds ◽  
Unicyclic Graphs ◽  
Degree Of A Vertex

The forgotten topological index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] overall edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. The graph [Formula: see text] is called a quasi-unicyclic graph if there exists a vertex [Formula: see text] such that [Formula: see text] is a connected graph with a unique cycle. In this paper, we give sharp upper and lower bounds for the F-index (forgotten topological index) of the quasi-unicyclic graphs.

scholarly journals A Note on the Estrada Index of the Aα-Matrix

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 811
Author(s):  
Jonnathan Rodríguez ◽  
Hans Nina
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Estrada Index

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A(G) and D(G) and defined the Aα-matrix for every real α∈[0,1] as: Aα(G)=αD(G)+(1−α)A(G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the Aα-matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

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