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.

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 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.

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.

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.

THE GUTMAN INDEX OF UNICYCLIC GRAPHS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250031 ◽  
Author(s):  
LIHUA FENG
Keyword(s):  
Connected Graph ◽  
Unicyclic Graphs ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Gutman Index

Let G be a connected graph with vertex set V(G). The Gutman index of G is defined as S(G) = ∑{u, v}⊆V(G) d(u)d(v)d(u, v), where d(u) is the degree of vertex u, and d(u, v) denotes the distance between u and v. In this paper, we characterize n-vertex unicyclic graphs with girth k, having minimal Gutman index.

scholarly journals The number of independent sets in unicyclic graphs

2005 ◽  
Vol 152 (1-3) ◽  
pp. 246-256 ◽  
Author(s):  
Anders Sune Pedersen ◽  
Preben Dahl Vestergaard
Keyword(s):  
Independent Sets ◽  
Unicyclic Graphs

scholarly journals Detour index of a class of unicyclic graphs

Filomat ◽  
2010 ◽  
Vol 24 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Qi Xuli ◽  
Bo Zhou
Keyword(s):  
Connected Graph ◽  
Unicyclic Graphs ◽  
The Third ◽  
Subject Classifications ◽  
Mathematics Subject Classifications

The detour index of a connected graph is defined as the sum of detour distances between all unordered pairs of vertices. We determine the n-vertex unicyclic graphs whose vertices on its unique cycle all have degree at least three with the first, the second and the third smallest and largest detour indices respectively for n ? 7. 2010 Mathematics Subject Classifications. 05C12, 05C35, 05C90. .

scholarly journals An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Muhammad Asif ◽  
Muhammad Hussain ◽  
Hamad Almohamedh ◽  
Khalid M. Alhamed ◽  
Sultan Almotairi
Keyword(s):  
Computer Program ◽  
Regular Graph ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Fixed Length ◽  
Inverse Degree ◽  
Fully Connected

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

scholarly journals Hamiltonian Cycles in Squares of Vertex-Unicyclic Graphs

1976 ◽  
Vol 19 (2) ◽  
pp. 169-172 ◽  
Author(s):  
Herbert Fleischner ◽  
Arthur M. Hobbs
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Cycles ◽  
Unicyclic Graphs ◽  
Necessary And Sufficient

In this paper we determine necessary and sufficient conditions for the square of a vertex-unicyclic graph to be Hamiltonian. The conditions are simple and easily checked. Further, we show that the square of a vertex-unicyclic graph is Hamiltonian if and only if it is vertex-pancyclic.

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