A topological index reflects the physical, chemical and structural properties of a molecule, and its study has an important role in molecular topology, chemical graph theory and mathematical chemistry. It is a natural problem to characterize non-isomorphic graphs with the same topological index value. By introducing a relation on trees with respect to edge division vectors, denoted by $\langle\mathcal{T}_n, \preceq \rangle$, in this paper we give some results for the relation order in $\langle\mathcal{T}_n, \preceq \rangle$, it allows us to compare the size of the topological index value without relying on the specific forms of them, and naturally we can determine which trees have the same topological index value. Based on these results we characterize some classes of trees that are uniquely determined by their edge division vectors and construct infinite classes of non-isomorphic trees with the same topological index value, particularly such trees of order no more than $10$ are completely determined.
We classify connected spanning convex subgraphs of the square cycles. We then show that every spanning tree of $C_n^2$ is contained in a unique nontrivial connected spanning convex subgraph of $C_n^2$. As a result, we obtain a purely combinatorial derivation of the formula for the number of spanning trees of the square cycles.
Let $G=(V,E)$ be a finite simple graph. The Graovac-Ghorbani index of a graph G is defined as $ABC_{GG}(G)=\sum_{uv\in E(G)}\sqrt{\frac{n_u(uv,G)+n_v(uv,G)-2}{n_u(uv,G)n_v(uv,G)}},$ where $n_u(uv,G)$ is the number of vertices closer to vertex $u$ than vertex $v$ of the edge $uv\in E(G)$. $n_v(uv,G)$ is defined analogously. The atom-bond connectivity index of a graph G is defined as $ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}},$ where $d_u$ is the degree of vertex $u$ in $G$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by select…
A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficientl…
We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is colored uniformly at random from the set $[n-1]$. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and $p = (2 + ε) \frac{\log n}{n}$ for some constant $ε> 0$, then $G_p$ almost surely contains a rainbow spanning tree. In this paper, we show that if $G$ is a $d$-regular $Ω(n)$-edge-connected graph, then when $p…
A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graph…
Let $G =(V_{G}, E_{G})$ be a simple connected graph with its vertex set $V_{G}$ and edge set $E_{G}$. The Mostar index $Mo(G)$ was defined as $Mo(G)=\sum\limits_{e=uv\in E(G)}|n_{u}-n_{v}|$, where $n_{u}$ (resp., $n_{v}$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion.
We show that for any fixed integer $m \geq 1$, a graph of maximum degree $Δ$ has a coloring with $O(Δ^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(Δ^{3/2})$ colors suffice, and acyclic colorings, for which $O(Δ^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph o…
This paper establishes sufficient conditions that force a graph to contain a bipartite subgraph with a given structural property. In particular, let $β$ be any of the following graph parameters: Hadwiger number, Hajós number, treewidth, pathwidth, and treedepth. In each case, we show that there exists a function $f$ such that every graph $G$ with $β(G)\geq f(k)$ contains a bipartite subgraph $\hat{G}\subseteq G$ with $β(\hat{G})\geq k$.
Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs.
