相关论文: No two starlikes have equal index

We say that an edge colouring breaks an automorphism if some edge is mapped to an edge of a different colour. We say that the colouring is distinguishing if it breaks every non-identity automorphism. We show that such colouring can be…

A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$…

The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \times \cdots \times n$ ($k$…

This paper investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalizes…

A k-ranking of a graph G is a labeling of the vertices of G with values from 1,...,k such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for…

An $e$-star is a complete bipartite graph $K_{1,e}$. An $e$-star system of order $n>1$, $S_e(n)$, is a partition of the edges of the complete graph $K_n$ into $e$-stars. An $e$-star system is said to be $k$-colourable if its vertex set can…

Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges,…

For a graph $G$, let $\lambda_2(G)$ denote the second largest eigenvalue of the adjacency matrix of $G$. We determine the extremal trees with maximum/minimum adjacency eigenvalue $\lambda_2$ in the class $\mathcal{T}(n,d)$ of $n$-vertex…

The Wiener index of a connected graph is the sum of topological distances between all pairs of vertices. Since Wang gave a mistake result on the maximum Wiener index for given tree degree sequence, in this paper, we investigate the maximum…

In this paper we develop three characterizations for isomorphism of graphs. The first characterization is obtained by associating certain bitableaux with the graphs. We order these bitableaux by suitably defined lexicographic order and…

(Im)balance indices can be used to quantify the (im)balance of trees by assigning numerical scores to them. An easy way to generate a new index is to construct a compound index, e.g., a linear combination of established indices. Two of the…

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ such that any nontrivial path whose endpoints have the same label contains a larger label. The least $k$ for which $G$ has a $k$-ranking is the ranking number…

In 2021, the Sombor index was introduced by Gutman, which is a new degree-based topological molecular descriptors. The Sombor index of a graph $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the…

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We…

A graph $G$ is asymmetrizable if it has a set of vertices whose setwise stablizer only consists of the identity automorphism. The motion $m$ of a graph is the minimum number of vertices moved by any non-identity automorphism. It is known…

We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue…

A labelling of a graph is an assignment of labels to its vertex or edge sets (or both), subject to certain conditions, a well established concept. A labelling of a graph G of order n is termed a numbering when the set of integers {1,...,n}…

A $k$-ranking of a directed graph $G$ is a labeling of the vertex set of $G$ with $k$ positive integers such that every directed path connecting two vertices with the same label includes a vertex with a larger label in between. The rank…

By weighted tree we understand such connected tree,that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we…

A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is…