The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.
Among many topological indices of trees the sum of distances $σ(T)$ and the number of subtrees $F(T)$ have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the "local" versions of these invariants, $σ_T(v)$ for the sum of distance from $v$ to all other vertices and $F_T(v)$ for the number of subtrees containing $v$, extremal problems can be raised and studied for vertices within a tree. This leads to th…
The \emph{blow-up} of a graph $H$ is the graph obtained from replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blow-ups of paths. We determined the extremal number and found the extremal graphs for the blow-ups of cycles and a large class of trees, when $n$ is sufficiently large. This generalizes their results. The additional aim of our note is to draw attention to a powerful tool…
The Randic (connectivity) index is one of the most successful molecular descriptors in structure-property and structure-activity relationships studies. J. Gao found the sharp upper bound for the Randic index of apex trees. In this paper, we proved that k-apex trees are not regular graphs for k\ge2 and proposed a sharp upper bound for the Randic index of k-apex trees for k>1.
Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The Sombor index of the graph $G$ is a degree-based topological index, defined as $$SO(G)=\sum_{uv \in E}\sqrt{d(u)^2+d(v)^2},$$ in which $d(x)$ is the degree of the vertex $x \in V$ for $x=u, v$. In this paper, we characterize the extremal trees with a given degree sequence that maximizes the Sombor index.
Vince and Wang showed that the average subtree density of a series-reduced tree is between 1/2 and 3/4, answering a conjecture of Jamison. They ask under what conditions a sequence of such trees may have average subtree density tending to either bound; we answer these questions by giving simple necessary and sufficient conditions in each case.
The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate achemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. The tree that minimizes the Wiener index among trees of given maximal degree was studied. We characterize trees that achieve the maximum and minimum Wiener index, given the number of vertices and the degree sequence.
We study the eigenvectors of Laplacian matrices of trees. The Laplacian matrix is reduced to a tridiagonal matrix using the Schur complement. This preserves the eigenvectors and allows us to provide fomulas for the ratio of eigenvector entries. We also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and Perron values. The results are then applied to the Fiedler vector. Here we locate the extremal entries of the Fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path.
The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and cha…
The first multiplicative Zagreb index $Π_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index $Π_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for $Π_1$ and upper bound for $Π_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds.
The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.
In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $π=(d_1, ..., d_5, 1,..., 1)$ have been characterized.
Let ${\mathcal T}(n,m)$ and ${\mathcal F}(n,m)$ denote the classes of weighted trees and forests, respectively, of order $n$ with the positive integral weights and the fixed total weight sum $m$, respectively. In this paper, we determine the minimum energies for both the classes ${\mathcal T}(n,m)$ and ${\mathcal F}(n,m)$. We also determine the maximum energy for the class ${\mathcal F}(n,m)$. In all cases, we characterize the weighted graphs whose energies reach these extremal values. We also solve the similar maximum energy and minimum energy problems for the classes of (0,1) weighted trees …
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).
Let G be a simple connected graph with n vertices, and let d_i be the degree of the vertex v_i in G. The extended adjacency matrix of G is defined so that the ij-entry is 1/2(d_i/d_j+d_j/d_i) if the vertices v_i and v_j are adjacent in G, and 0 otherwise. This matrix was originally introduced for developing novel topological indices used in the QSPR/QSAR studies. In this paper, we consider extremal problems of the largest eigenvalue of the extended adjacency matrix (also known as the extended spectral radius) of trees. We show that among all trees of order n>= 5, the path Pn(resp., the star Sn…
A new geometric background of graph invariants was introduced by Gutman, of which the simplest is the second Sombor index $SO_2$, defined as $SO_2=SO_2(G)=\sum_{uv\in E}\frac{|d^2_G(u)-d^2_G(v)|}{d^2_G(u)+d^2_G(v)}$, where $G = (V, E)$ is a simple graph and $d_G(v)$ denotes the degree of $v$ in $G$. In this paper, the chemical applicability of the second Sombor index is investigated and it is shown that the the second Sombor index is useful in predicting physicochemical properties with high accuracy compared to some well-established and often used indices. Also, we obtain a bound for the secon…
The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph, which is proposed first by Gutman and Wagner [The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177--2187]. And they gave some properties and asymptotic results of the matching energy. In this paper, we characterize the trees with $n$ vertices whose complements have the maximal, second-maximal and minimal matching energy. Further, we determine the trees with a perfect matching whose complements have the second-maximal matching energy. In particular, show that the …
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in applications. The number of subtrees in the extremal cases constitute sequences which are of interest to number theorists. The structures which maximize or minimize the number of subtrees among general trees, binary trees and trees with a given maximum degree have been identified previously. Most recently, results of this nature are generalized to trees with a giv…
The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ are linear in $n$, the latter posed by Braß-Károlyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in …
The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If $G=(V_G,E_G)$ is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of $G$ is defined as $ξ^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$. In this paper we first introduced four edge-grafting transformations to study the mathematical p…
Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $ξ^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) = \sum_{u\in V_G}d_G(u,v)$ is the sum of all distances from the vertex $v$. In this paper the tree among $n$-vertex trees with domination number $γ$ having the minimal eccentric distance sum is determined and the tree among $n$-vertex trees with domination number $γ$ satisfying $n = kγ$ having the maximal eccentric distance sum is identified, respectively, for $k=2,3,\fr…
A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives a distinct color. In this paper we consider the natural extremal problem of maximizing and minimizing the number of rainbow spanning trees in a graph $G$. Such a question clearly needs restrictions on the colorings to be meaningful. For edge-colorings using $n-1$ colors and without rainbow cycles, known in the literature as JL-colorings, there turns out to be a particularly nice way of counting the rainbow spanning trees and we solve this problem completely for JL-colored complete graphs $K_n$ and comp…
The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. %Some general properties of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set of all $n$-vertex trees, $\mathbb{T}_{n,Δ}$ be the set of $n$-vertex trees with given maximum degree equal to $Δ$, $\mathbb{T}_n^k$ be the set of $n$-vertex trees with exactly $k$ vertices of maximum degree, and let $\mathbb{T}_{n,Δ}^k$ be the set of $n$-vertex trees with exactly $k$ vertices of given maximum degree equal to $Δ.$ In this paper, we first determine…
Recently, D. Vuki$\check{c}$evi$\acute{c}$ and J. Sedlar in \cite{Vuki} introduced an order "$\preceq$" on $\mathcal{T}_n$, the set of trees on $n$ vertices, such that the topological index $F$ of a graph is a function defined on the order set $\langle\mathcal{T}_n,\preceq\rangle$. It provides a new approach to determine the extremal graphs with respect to topological index $F$. By using the method they determined the common maximum and/or minimum graphs of $\mathcal{T}_n$ with respect to topological indices of Wiener type and anti-Wiener type. Motivated by their researches we further study th…
Let $\mathcal{CT}_{n,k}$ and $\mathcal{CT}^*_{n,b}$ be the classes of all $n$-vertex chemical trees with $k$ segments and $b$ branching vertices, respectively, where $3\le k\le n-1$ and $1\le b< \frac{n}{2}-1$. The solution of the problem of finding trees from the class $\mathcal{CT}_{n,k}$ or $\mathcal{CT}^*_{n,b}$, with the minimum first Zagreb index or minimum second Zagreb index follows directly from the main results of [MATCH Commun. Math. Comput. Chem. 72 (2014) 825-834] or [MATCH Commun. Math. Comput. Chem. 74 (2015) 57-79]. In this paper, the chemical trees with the maximum first/secon…
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.
For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index $\prod_2$ is equal to the product of the products of degrees of pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajstić in 1972. In this paper, we explore the mutiplicative Zagreb indices in terms of arbitrary domination number. The sharp upper and lower bounds of $\prod_1(G)$ and $\prod_2(G)$ are given. In addition,…
Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph $G$ as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}=d_{G}(u)$ denotes the degree of vertex $u$ in graph $G$. Extremal problems on the Sombor index for trees with a given diameter has been considered by Chen et al. [H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022) 23--49] and Li et al. [S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter, Appl. Math. Comput. 416 (2022) 126…
The problem of determining the largest number $f(n;\barκ\leq \ell)$ of edges for graphs with $n$ vertices and maximal local connectivity at most $\ell$ was considered by Bollobás. Li et al. studied the largest number $f(n;\barκ_3\leq2)$ of edges for graphs with $n$ vertices and at most two internally disjoint Steiner trees connecting any three vertices. In this paper, we further study the largest number $f(n;\barκ_k=1)$ of edges for graphs with $n$ vertices and exactly one Steiner tree connecting any $k$ vertices with $k\geq 3$. It turns out that this is not an easy task to finish, not like th…
The Wiener polarity index $W_p$, one of the most studied molecular structure descriptors, was devised by the chemist Harold Wiener for predicting the boiling points of alkanes. The index $W_p$ for chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices at distance 3. A vertex of a chemical tree with the degree at least 3 is called a branching vertex. A segment of a chemical tree $T$ is a path-subtree $S$ whose terminal vertices have degrees different from 2 in $T$ and every internal vertex (if exists) of $S$ has degree 2 in $T$. In this pap…
In a graph $G$, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of $G$, denoted by $ψ(G)$, is the cardinality of a maximum dissociation set of $G$. Extremal problems involving counting the number of a given type of substructure in a graph have been a hot topic of study in extremal graph theory throughout the last few decades. In this paper, we determine the maximum number of maximum dissociation sets in a tree with prescribed dissociation number and…
The problem of determining the smallest number of edges, $h(n;\barκ\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\barκ\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erdös and Gallai. Bollobás considered the problem of determining the largest number of edges $f(n;\barκ\leq \ell)$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\barκ\leq \ell)= h(n;\barκ\geq \ell+1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above prob…
The Harary index of a graph $G$ is recently introduced topological index, defined on the reverse distance matrix as $H(G)=\sum_{u,v \in V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the length of the shortest path between two distinct vertices $u$ and $v$. We present the partial ordering of starlike trees based on the Harary index and we describe the trees with the second maximal and the second minimal Harary index. In this paper, we investigate the Harary index of trees with $k$ pendent vertices and determine the extremal trees with maximal Harary index. We also characterize the extremal trees wi…
In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$ denote the family of $d$--regular graphs. The general problem studied in this paper is to determine $$\inf_{G\in \mathcal{G}_d}\hom(G,A)^{1/v(G)}.$$ It turns out that in many instances the infimum is not achieved by a finite graph, but a sequence of graphs with girth (i. e., length of the shortest cycle) tending to infinity. In other words, the optimization pr…
For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of $H$-colorings. In this paper, we prove several results in this area. First, we find a class of graphs ${\mathcal H}$ with the property that for each $H \in {\mathcal H}$, the $n$-vertex tree that minimizes the number of $H$-colorings is …
The terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arises in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents a sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees which attain these bounds. In addition, we investigate the properties of extremal trees which attain the maximum terminal Wiener index among all trees of order $n$ with fixed maximum degree.
We study that over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees. Trees minimizing (resp. maximizing) the total number of subtrees usually maximize (resp. minimize) the Wiener index, and vice versa. Here are some of our results: (1) Let $\mathscr{T}_n^k$ be the set of all $n$-vertex trees with $k$ leaves, we determine the maximum (resp. minimum) value of the total number of subtrees of trees among $\mathscr{T}_n^k$ and characterize the extremal graphs. (2) Let $\mathscr{P}_n^{p,q}$ be the set of all $n$-vertex trees, each of …
We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics "DEZ" and "lec". Explicit formulas for the maximal case are derived by using symmetric function tools.
The connective eccentricity index $ξ^{ce}=\sum^{}_{u\in V}\frac{d(u)}{\varepsilon(u)}$, where $\varepsilon(u)$ and $d(u)$ denote the eccentricity and the degree of the vertex $u$, respectively. In this paper, we first determine the extremal trees which minimize and maximize the connective eccentricity index among all trees with a given degree sequence, and then determine the extremal trees which minimize and maximize the connective eccentricity index among all trees with a given number of branching vertices.
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 the recent results of Kirk and Wang. These trees coincide with those which were proven by Wang and independently Zhang et al. to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.
The eccentricity of a vertex, $ecc_T(v) = \max_{u\in T} d_T(v,u)$, was one of the first, distance-based, tree invariants studied. The total eccentricity of a tree, $Ecc(T)$, is the sum of eccentricities of its vertices. We determine extremal values and characterize extremal tree structures for the ratios $Ecc(T)/ecc_T(u)$, $Ecc(T)/ecc_T(v)$, $ecc_T(u)/ecc_T(v)$, and $ecc_T(u)/ecc_T(w)$ where $u,w$ are leaves of $T$ and $v$ is in the center of $T$. In addition, we determine the tree structures that minimize and maximize total eccentricity among trees with a given degree sequence.
Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection ($TBR$), subtree prune and regraft ($SPR$) and rooted subtree prune and regraft ($rSPR$). For a pair of leaf-labelled binary trees with $n$ leaves, the maximum number of such moves required to transform one into the other is $n-Θ(\sqrt{n})$, extending a result of Ding, Grunewald and Humphries. We show that if the pair is chosen uniformly at random, then the expected number of moves required to transfer one into the other is $n-Θ(n^{2/3})$. These results m…
The concept of maximum local connectivity $\bar κ$ of a graph was introduced by Bollobás. One of the problems about it is to determine the largest number of edges $f(n;\barκ\leq \ell)$ for graphs of order $n$ that have local connectivity at most $\ell$. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{generalized local connectivity} $κ(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\barκ_k(G)=max\{κ(S)|S\subseteq V(G),|S|=k\}$ is called the \emph{maximum generalized local connectivity} …
Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. In this paper, we present a sharp upper bound for the signless spectral radius of $G$ without any tree and characterize all extremal graphs which attain the upper bound, which may be regarded as a spectral extremal version for the famous Erdős-Sós conjecture.
We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We completely solve this problem when $H$ is a clique, as well as more generally when $H$ is any regular graph with degree at least about half its number of vertices. We also characterize the extremal graphs when $H$ is an Erdős-Rényi random graph. The extremal structures turn out to have the similar form as the conjectured extremal structures for a well-studied b…
In this paper we present a novel approach in extremal set theory which may be viewed as an asymmetric version of Katona's permutation method. We use it to find more Turán numbers of hypergraphs in the Erdős--Ko--Rado range. An $(a,b)$-path $P$ of length $2k-1$ consists of $2k-1$ sets of size $r=a+b$ as follows. Take $k$ pairwise disjoint $a$-element sets $A_0, A_2, \dots, A_{2k-2}$ and other $k$ pairwise disjoint $b$-element sets $B_1, B_3, \dots, B_{2k-1}$ and order them linearly as $A_0, B_1, A_2, B_3, A_4\dots$. Define the (hyper)edges of $P_{2k-1}(a,b)$ as the sets of the form $A_i\cup B_{…
