Let $G$ be a connected graph of order $n\ge 3$ and let $cE\left(G\right)\to \{1,2,...,k\}$ be a coloring of the edges of $G$ (where adjacent edges may be colored the same). For each vertex $v$ of $G$, the color code of $v$ with respect to $c$ is the $k$-tuple $c\left(v\right)=(a_1,a_2,\cdots ,a_k)$, where $a_i$ is the number of edges incident with $v$ that are colored $i$ ($1\le i\le k$). The coloring $c$ is detectable if distinct vertices have distinct color codes. The detection number $det\left(G\right)$ of $G$ is the minimum positive integer $k$ for which $G$ has a detectable $k$-coloring. We establish a formula for the detection...

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues ${\rho }_1^L\ge {\rho }_2^L\ge \cdots \ge {\rho }_n^L$, the distance Laplacian energy $\mathrm{DLE}\left(G\right)$ is defined as $\mathrm{DLE}\left(G\right)={\sum }_{i=1}^n|{\rho }_i^L-2W\left(G\right)/n|$, where $W\left(G\right)$ is the Wiener index of $G$. We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $\mathrm{DLE}\left(G\right)$ in terms of the order $n$, the Wiener index $W\left(G\right)$, the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs...

In this paper, we give some sufficient conditions for distance local connectivity of a graph, and a degree condition for local connectivity of a k-connected graph with large diameter. We study some relationships between t-distance chromatic number and distance local connectivity of a graph and give an upper bound on the t-distance chromatic number of a k-connected graph with diameter d.

Discrete partially ordered sets can be turned into distance spaces in several ways. The distance functions may or may not satisfy the triangle inequality and restrictions of the distance to finite chains may or may not coincide with the natural, difference-of-height distance measured in a chain. It is shown that for semilattices the semimodularity ensures the good behaviour of the distances considered. The Jordan-Dedekind chain condition, which is weaker than semimodularity, is equivalent to the...

In 1986, Chartrand, Saba and Zou [3] defined a measure of the distance between (the isomorphism classes of) two graphs based on 'edge rotations'. Here, that measure and two related measures are explored. Various bounds, exact values for classes of graphs and relationships are proved, and the three measures are shown to be intimately linked to 'slowly-changing' parameters.

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number ${\chi }_D\left(G\right)$ of G. Extending these concepts to infinite graphs we prove that $D\left(Q_{\aleph }₀\right)=2$ and ${\chi }_D\left(Q_{\aleph }₀\right)=3$, where $Q_{\aleph }₀$ denotes the hypercube of countable dimension. We also show that ${\chi }_D\left(Q₄\right)=4$, thereby completing the investigation of finite hypercubes with respect to ${\chi }_D$. Our results...