scholarly journals The minimal total irregularity of some classes of graphs

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1203-1211 ◽  
Author(s):  
Yingxue Zhu ◽  
Lihua You ◽  
Jieshan Yang
Keyword(s):  
Open Problem ◽  
Vertex Degree ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
The Third ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

In [1], Abdo and Dimitov defined the total irregularity of a graph G=(V,E) as irrt(G)=1/2 ?u,v?V|dG(u)-dG(v)|, where dG(u) denotes the vertex degree of a vertex u ? V. In this paper, we investigate the minimal total irregularity of the connected graphs, determine the minimal, the second minimal, the third minimal total irregularity of trees, unicyclic graphs, bicyclic graphs on n vertices, and propose an open problem for further research.

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).

scholarly journals The Entropy of Weighted Graphs with Atomic Bond Connectivity Edge Weights

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Young Chel Kwun ◽  
Hafiz Mutee ur Rehman ◽  
Muhammad Yousaf ◽  
Waqas Nazeer ◽  
Shin Min Kang
Keyword(s):  
Open Problem ◽  
Bipartite Graphs ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Star Graphs ◽  
Chemical Graphs ◽  
Complete Bipartite Graphs

The aim of this report to solve the open problem suggested by Chen et al. We study the graph entropy with ABC edge weights and present bounds of it for connected graphs, regular graphs, complete bipartite graphs, chemical graphs, tree, unicyclic graphs, and star graphs. Moreover, we compute the graph entropy for some families of dendrimers.

scholarly journals The Orderings of Bicyclic Graphs and Connected Graphs by Algebraic Connectivity

10.37236/434 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianxi Li ◽  
Ji-Ming Guo ◽  
Wai Chee Shiu
Keyword(s):  
Linear Algebra ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Smallest Eigenvalue ◽  
Bicyclic Graphs

The algebraic connectivity of a graph $G$ is the second smallest eigenvalue of its Laplacian matrix. Let $\mathscr{B}_n$ be the set of all bicyclic graphs of order $n$. In this paper, we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in $\mathscr{B}_n$ when $n\geq 13$. This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order $n$. This extends the results of Shao et al. [The ordering of trees and connected graphs by their algebraic connectivity, Linear Algebra Appl. 428 (2008) 1421-1438].

scholarly journals More results on extremum Randic indices of (molecular) trees

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3581-3590 ◽  
Author(s):  
Nor Husin ◽  
Roslan Hasni ◽  
Zhibin Du ◽  
Akbar Ali
Keyword(s):  
Randić Index ◽  
Unicyclic Graphs ◽  
Randic Index ◽  
The Third ◽  
Degree Of Vertex ◽  
Bicyclic Graphs ◽  
Molecular Trees

The Randic index R(G) of a graph G is the sum of the weights (dudv)-1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randic indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex trees with the third for n ? 7, the fourth for n ? 10, the fifth and the sixth for n ? 11 maximum Randic indices. Recently, Li et al. [The Randic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex trees with the seventh, the eighth, the ninth and the tenth for n ? 11 maximum Randic indices. In this paper, we correct the ordering for the Randic indices of trees obtained by Li et al., and characterize the trees with from the seventh to the sixteenth maximum Randic indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.

scholarly journals The Connected Detour Numbers of Special Classes of Connected Graphs

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Ahmed M. Ali ◽  
Ali A. Ali
Keyword(s):  
Minimum Order ◽  
Induced Subgraph ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Bicyclic Graphs ◽  
Special Classes

Simple finite connected graphs G=V,E of p≥2 vertices are considered in this paper. A connected detour set of G is defined as a subset S⊆V such that the induced subgraph GS is connected and every vertex of G lies on a u−v detour for some u,v∈S. The connected detour number cdnG of a graph G is the minimum order of the connected detour sets of G. In this paper, we determined cdnG for three special classes of graphs G, namely, unicyclic graphs, bicyclic graphs, and cog-graphs for Cp, Kp, and Km,n.

scholarly journals NONREGULAR GRAPHS WITH MINIMAL TOTAL IRREGULARITY

2015 ◽  
Vol 92 (1) ◽  
pp. 1-10 ◽  
Author(s):  
HOSAM ABDO ◽  
DARKO DIMITROV
Keyword(s):  
Lower Bound ◽  
Undirected Graph ◽  
Connected Graphs ◽  
The Third ◽  
Odd Order ◽  
Degree Of A Vertex ◽  
Even Order ◽  
Sharp Lower Bound

The total irregularity of a simple undirected graph $G$ is defined as $\text{irr}_{t}(G)=\frac{1}{2}\sum _{u,v\in V(G)}|d_{G}(u)-d_{G}(v)|$, where $d_{G}(u)$ denotes the degree of a vertex $u\in V(G)$. Obviously, $\text{irr}_{t}(G)=0$ if and only if $G$ is regular. Here, we characterise the nonregular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu et al. [‘The minimal total irregularity of graphs’, Preprint, 2014, arXiv:1404.0931v1 ] about the lower bound on the minimal total irregularity of nonregular connected graphs. We show that the conjectured lower bound of $2n-4$ is attained only if nonregular connected graphs of even order are considered, while the sharp lower bound of $n-1$ is attained by graphs of odd order. We also characterise the nonregular graphs with the second and the third smallest total irregularity.

scholarly journals The Maximal Total Irregularity of Bicyclic Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lihua You ◽  
Jieshan Yang ◽  
Yingxue Zhu ◽  
Zhifu You
Keyword(s):  
Vertex Degree ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

In 2012, Abdo and Dimitrov defined the total irregularity of a graphG=(V,E)asirrtG=1/2∑u,v∈VdGu-dGv, wheredGudenotes the vertex degree of a vertexu∈V. In this paper, we investigate the total irregularity of bicyclic graphs and characterize the graph with the maximal total irregularity among all bicyclic graphs onnvertices.

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

Sharp upper bounds of F-index among bicyclic graphs

2021 ◽  
pp. 2250055
Author(s):  
R. Khoeilar ◽  
A. Jahanbani ◽  
L. Shahbazi ◽  
J. Rodríguez
Keyword(s):  
Maximum Degree ◽  
Topological Index ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

The [Formula: see text]-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]. In this paper, we give sharp upper bounds of the [Formula: see text]-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

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