The central vertices and radius of the regular graph of ideals
Farzad
Shaveisi
Razi University
author
text
article
2017
eng
The regular graph of ideals of the commutative ring $R$, denoted by ${\Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is proved that the radius of $\Gamma_{reg}(R)$ equals $3$. The central vertices of $\Gamma_{reg}(R)$ are determined, too.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
4
no.
2017
1
13
http://toc.ui.ac.ir/article_21472_57a7aea214c4516a524744b78f00943a.pdf
dx.doi.org/10.22108/toc.2017.21472
The harmonic index of subdivision graphs
Bibi Naimeh
Onagh
Golestan University
author
text
article
2017
eng
The harmonic index of a graph $G$ is defined as the sum of the weights $\frac{2}{\deg_G(u)+\deg_G(v)}$ of all edges $uv$ of $G$, where $\deg_G(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we study the harmonic index of subdivision graphs, $t$-subdivision graphs and also, $S$-sum and $S_t$-sum of graphs.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
4
no.
2017
15
27
http://toc.ui.ac.ir/article_21471_6d4574ac2fe03052a0872fb991c96309.pdf
dx.doi.org/10.22108/toc.2017.21471
Splices, Links, and their Edge-Degree Distances
Mahdieh
Azari
Kazerun Branch, Islamic Azad University
author
Hojjatollah
Divanpour
Shiraz Technical College, Technical and Vocational University
author
text
article
2017
eng
The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)+d(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
4
no.
2017
29
42
http://toc.ui.ac.ir/article_21614_033f4714ff9a47c358a450a46e9a3122.pdf
dx.doi.org/10.22108/toc.2017.21614
On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph
Hanyuan
Deng
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
author
S.
Balachandran
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
author
S. K.
Ayyaswamy
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
author
Y. B.
Venkatakrishnan
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
author
text
article
2017
eng
The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc\left(G\right)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The harmonic index $H\left(G\right)$ of a graph $G$ is defined as the sum of $\frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of $G$, where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of $n$-vertex trees with two adjacent vertices of maximum degree $\Delta$, where $n\geq 2\Delta$. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi\'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas \cite{hl}.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
4
no.
2017
43
50
http://toc.ui.ac.ir/article_21470_6107bccf810358fdefb9471c7d0ba0a8.pdf
dx.doi.org/10.22108/toc.2017.21470
Some topological indices and graph properties
Xiaomin
Zhu
College of Science, Nantong University, Nantong, China
author
Lihua
Feng
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
author
Minmin
Liu
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
author
Weijun
Liu
School of Science, Nantong University, Nantong,226019， China,
author
Yuqin
Hu
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
author
text
article
2017
eng
In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected or $k$-connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.
Transactions on Combinatorics
University of Isfahan
2251-8657
6
v.
4
no.
2017
51
65
http://toc.ui.ac.ir/article_21467_d05e2410fc3d5c5560f3430866b8af0e.pdf
dx.doi.org/10.22108/toc.2017.21467