The largest Wiener index of unicyclic graphs given girth or maximum degree

2015 ◽  
Vol 53 (1-2) ◽  
pp. 343-363 ◽  
Author(s):  
Shang-wang Tan ◽  
Yan Lin
Keyword(s):  
Maximum Degree ◽  
Wiener Index ◽  
Unicyclic Graphs

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

2021 ◽  
pp. 2250069
Author(s):  
Shamaila Yousaf ◽  
Akhlaq Ahmad Bhatti
Keyword(s):  
Maximum Degree ◽  
Discrete Math ◽  
Simple Approach ◽  
Unicyclic Graphs ◽  
Irregularity Index ◽  
Degree Of A Vertex ◽  
Tricyclic Graphs ◽  
Bicyclic Graphs ◽  
Appl Math

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).

Extremal trees with respect to the Steiner Wiener index

2019 ◽  
Vol 11 (06) ◽  
pp. 1950067
Author(s):  
Jie Zhang ◽  
Guang-Jun Zhang ◽  
Hua Wang ◽  
Xiao-Dong Zhang
Keyword(s):  
Maximum Degree ◽  
Computational Analysis ◽  
Degree Sequence ◽  
Wiener Index ◽  
Extremal Problems ◽  
Difficult Problem ◽  
Computational Results ◽  
Number Of Leaves ◽  
Extremal Structure ◽  
Chemical Trees

The well-known Wiener index is defined as the sum of pairwise distances between vertices. Extremal problems with respect to it have been extensively studied for trees. A generalization of the Wiener index, called the Steiner Wiener index, takes the sum of minimum sizes of subgraphs that span [Formula: see text] given vertices over all possible choices of the [Formula: see text] vertices. We consider the extremal problems with respect to the Steiner Wiener index among trees of a given degree sequence. First, it is pointed out minimizing the Steiner Wiener index in general may be a difficult problem, although the extremal structure may very likely be the same as that for the regular Wiener index. We then consider the upper bound of the general Steiner Wiener index among trees of a given degree sequence and study the corresponding extremal trees. With these findings, some further discussion and computational analysis are presented for chemical trees. We also propose a conjecture based on the computational results. In addition, we identify the extremal trees that maximize the Steiner Wiener index among trees with a given maximum degree or number of leaves.

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.

scholarly journals Differentials in certain classes of graphs

2010 ◽  
Vol 41 (2) ◽  
pp. 129-138 ◽  
Author(s):  
P. Roushini Leely Pushpam ◽  
D. Yokesh
Keyword(s):  
Binary Tree ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Regular Graphs ◽  
Complete Binary Tree ◽  
Unicyclic Graphs ◽  
Grid Graphs ◽  
Split Graphs ◽  
Delta G

Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

scholarly journals Unicyclic Graphs with the Fourth Extremal Wiener Indices

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Guangfu Wang ◽  
Yujun Yang ◽  
Yuliang Cao ◽  
Shoujun Xu
Keyword(s):  
Wiener Index ◽  
Unicyclic Graphs

A graph is called unicyclic if the graph contains exactly one cycle. Unicyclic graphs with the fourth extremal Wiener indices are characterized. It is shown that, among all unicyclic graphs with n≥8 vertices, C5Sn−4 and C2u1,u2S3,Sn−4 attain the fourth minimum Wiener index, whereas C3u1,u2P3,Pn−4 attains the fourth maximum Wiener index.

The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices

2016 ◽  
Vol 55 (1-2) ◽  
pp. 1-24 ◽  
Author(s):  
Shang-wang Tan ◽  
Qi-long Wang ◽  
Yan Lin
Keyword(s):  
Wiener Index ◽  
Unicyclic Graphs

Remarks on the Wiener index of unicyclic graphs

2012 ◽  
Vol 41 (1-2) ◽  
pp. 49-59 ◽  
Author(s):  
R. Nasiri ◽  
H. Yousefi-Azari ◽  
M. R. Darafsheh ◽  
A. R. Ashrafi
Keyword(s):  
Wiener Index ◽  
Unicyclic Graphs

scholarly journals Wiener index in graphs with given minimum degree and maximum degree

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Peter Dankelmann ◽  
Alex Alochukwu
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Large Degree ◽  
Wiener Index ◽  
Sharp Bound

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.

scholarly journals The Variable Wiener Index of Trees with Given Maximum Degree

2014 ◽  
Author(s):  
Zhen Jia ◽  
Ning Lin ◽  
Hongjie Gao ◽  
Zhilin Chen ◽  
Zhiyuan Wang ◽  
...  
Keyword(s):  
Maximum Degree ◽  
Wiener Index

The harmonic index of unicyclic graphs

2017 ◽  
Vol 10 (03) ◽  
pp. 1750039 ◽  
Author(s):  
R. Rasi ◽  
S. M. Sheikholeslami
Keyword(s):  
Maximum Degree ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Harmonic Index ◽  
Degree Of A Vertex

The harmonic index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. Hu and Zhou [WSEAS Trans. Math. 12 (2013) 716–726] proved that for any unicyclic graph [Formula: see text] of order [Formula: see text], [Formula: see text] with equality if and only if [Formula: see text]. Recently, Zhong and Cui [Filomat 29 (2015) 673–686] generalized the above bound and proved that for any unicyclic graph [Formula: see text] of order [Formula: see text] other than [Formula: see text], [Formula: see text]. In this paper, we generalize the aforemention results and show that for any connected unicyclic graph [Formula: see text] of order [Formula: see text] with maximum degree [Formula: see text], [Formula: see text] and classify the extremal unicyclic graphs.

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