arXiv subjects

Kexiang Xu

Publications and source records attributed to Kexiang Xu.

23 records · Page 2Linked to original sources

arXiv2020

Constructing new families of transmission irregular graphs

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree $T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$ pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if a starlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it is transmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmission irregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. Additional infinite families among the starlike trees and bi-starlike trees are determined. Transmission irregular unicyclic infinite families are also presented, in particular, the line graph of $T(a,a+1,a+2)$, $a\ge 2$, is transmission irregular if and only if $a$ is even.

math.CO↗

arXiv2019

Comparison of Wiener index and Zagreb eccentricity indices

The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ is the eccentricity of a vertex $v$. In this paper the invariants $E_1$, $E_2$, and the Wiener index are compared on graphs with diameter $2$, on trees, on a newly introduced class of universally diametrical graphs, and on Cartesian product graphs. In particular, if the diameter of a tree $T$ is not too big, then $W(T) \ge E_2(T)$ holds, and if the diameter of $T$ is large, then $W(T) < E_1(T)$ holds.

math.CO↗

arXiv2018

Extremal graphs for vertex-degree-based invariants with given degree sequences

For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $π=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invariants based on degree are included.

math.CO↗

arXiv2017

Embeddings into almost self-centered graphs of given radius

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius $r$ is called an $r$-ASC graph. The $r$-ASC index $θ_r(G)$ of a graph $G$ is the minimum number of vertices needed to be added to $G$ such that an $r$-ASC graph is obtained that contains $G$ as an induced subgraph. It is proved that $θ_r(G)\le 2r$ holds for any graph $G$ and any $r\ge 2$ which improves the earlier known bound $θ_r(G)\le 2r+1$. It is further proved that $θ_r(G)\le 2r-1$ holds if $r\geq 3$ and $G$ is of order at least $2$. The $3$-ASC index of complete graphs is determined. It is proved that $θ_3(G)\in \{3,4\}$ if $G$ has diameter $2$ and for several classes of graphs of diameter $2$ the exact value of the $3$-ASC index is obtained. For instance, if a graph $G$ of diameter $2$ does not contain a diametrical triple, then $θ_3(G) = 4$. The $3$-ASC index of paths of order $n\geq 1$, cycles of order $n\geq 3$, and trees of order $n\geq 10$ and diameter $n-2$ are also determined, respectively, and several open problems proposed.

math.CO↗

arXiv2016

Ordering connected graphs by their Kirchhoff indices

The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distances between all unordered pairs of vertices, which was introduced by Klein and Randić. In this paper we characterized all extremal graphs with Kirchhoff index among all graphs obtained by deleting $p$ edges from a complete graph $K_n$ with $p\leq\lfloor\frac{n}{2}\rfloor$ and obtained a sharp upper bound on the Kirchhoff index of these graphs. In addition, all the graphs with the first to ninth maximal Kirchhoff indices are completely determined among all connected graphs of order $n>27$.

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