Pavel Hrnčiar
Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We show that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$. Moreover, for all positive integers $r$ and $d$ with $r\leq d\leq 2r-2$ there exists a graph of radius $r$ and diameter $d$ with circumference $4r-2d$.
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Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.
The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on one fixed geodesic between a pair of vertices from $S$. In this paper, some general properties of the strong geodesic problem are studied, especially in connection with diameter of a graph. The problem is also solved for balanced complete bipartite graphs.
For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely characterized.
The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.
We study the Randic index for cactus graphs. It is conjectured to be bounded below by radius (for other than an even path), and it is known to obey several bounds based on diameter. We study radius and diameter for cacti then verify the radius bound and strengthen two diameter bounds for cacti. Along the way, we produce several other bounds for the Randic index in terms of graph size, order, and valency for several special classes of graphs, including chemical nontrivial cacti and cacti with starlike BC-trees.
Among other things, it is shown that for every pair of positive integers $r$, $d$, satisfying $1<r<d\leq 2r$, and every finite simple graph $H,$ there is a connected graph $G$ with diameter $d$, radius $r$, and center $H.$
Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of natural open problems.
Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join of a complete graph of order $k$ and an empty graph of order $n-k$. In this paper, we show that the conjecture is true for trees of diameter at most four.
The diameter and radius of radially maximal graphs
本文证明了Dutton等人关于径向极大图存在性的猜想：对于满足$r\le d\le 2r-2$的任意正整数$r$和$d$，存在半径为$r$、直径为$d$的径向极大图，并推广了该结果。
Let $S_{1}(m, d, k)$ be the $k$-uniform supertree obtained from a loose path $P:v_{1}, e_{1}, v_{2}, \ldots,v_{d}, e_{d}, v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{\lfloor\frac{d}{2}\rfloor+1}.$ Let $\mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $S_{1}(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $\mathbb{S}(m,d,k)$ for $3\leq d\leq m-1$. Furthermore, we determine the uniqu…
We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter.
The spectral radius $ρ(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph $G$ with $ρ(G)\leq 3/2{\sqrt{2}}$ is either a dagger, an open quipu, or a closed quipu. The reverse statement is not true. Many open quipus and closed quipus have spectral radius greater than $3/2{\sqrt{2}}$. In this paper we proved the following results. For any open quipu $G$ on $n$ vertices ($n\geq 6$) with spectral radius less than $3/2{\sqrt{2}}$, its diameter $D(G)$ satisfies $D(G)\geq (2n-4)/3$. This bound is tight. For any closed quipu $G$ on $…
Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (λI - L (G))=\sum_{k = 0}^n (-1)^k c_k λ^{n - k}$. It is well known that for trees the Laplacian coefficient $c_{n-2}$ is equal to the Wiener index of $G$. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize first the trees with given diameter and then the connected graphs with given radius which simultaneously minimize all Laplacian coefficients. This approach generaliz…
Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_α(G) = αD(G) + (1-α)A(G)$ for any real $α\in[0,1]$. The largest eigenvalue of $A_α(G)$ is called the $A_α$ spectral radius or the $A_α$-index of $G$. Let $\mathcal{G}_{n,k}^d$ be the set of $k$-connected graphs of order $n$ with diameter $d$. In this paper, we determine the graphs with maximum $A_α$ spectral radius among all graphs in $\mathcal{G}_{n,k}^d$ for any $α\in[0,1)$, where $k\geq2$ and $d\geq2$. We generalizes the results about adjacency matrix of Theorem 3.6 i…
The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. A minimizer graph is such that minimizes the spectral radius among all connected graphs on $n$ vertices with diameter $d$. The minimizer graphs are known for $d\in\{1,2\}\cup [n/2,2n/3-1]\cup\{n-k\mid k=1,2,...,8\}$. In this paper, we determine all minimizer graphs for $d=2(n-1)/3$.
Let $μ(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. Let $S_{n,k}=K_{k}\vee\overline{K_{n-k}}$, and let $S_{n,k}^{+}$ be the graph obtained from $S_{n,k}$ by adding a single edge joining two vertices of the independent set of $S_{n,k}$. In 2010, Nikiforov conjectured that for a given integer $k$, every graph $G$ of sufficiently large order $n$ with $μ(G)\geq μ(S_{n,k}^{+})$ contains …
The spectral radius $ρ(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \sim 1 \sim 2 \sim ... \sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\in\{1,2,...,t\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioabǎ-van Dam-Koolen-Lee \cite{CDK} conjectured for fixed $e\…
Given a connected graph $G=(V,E)$ and a vertex set $S\subset V$, the {\em Steiner distance} $d(S)$ of $S$ is the size of a minimum spanning tree of $S$ in $G$. For a connected graph $G$ of order $n$ and an integer $k$ with $2\leq k \leq n$, the $k$-eccentricity of a vertex $v$ in $G$ is the maximum value of $d(S)$ over all $S\subset V$ with $|S|=k$ and $v\in S$. The minimum $k$-eccentricity, ${srad}_k(G)$, is called the $k$-radius of $G$ while the maximum $k$-eccentricity, ${sdiam}_k(G)$, is called the $k$-diameter of $G$. In 1990, Henning, Oellermann, and Swart [\textit{Ars Combinatoria} \tex…
Block eccentricity, a radius bound, and an application to the Randić index
本文提出基于块的偏心率框架，将半径与中心概念扩展到块，据此给出直径的新下界，并验证仙人掌图满足顶点半径与Randić指标之间的猜想界。
Median graphs form the class of graphs which is the most studied in metric graph theory. Recently, Bénéteau et al. [2019] designed a linear-time algorithm computing both the $Θ$-classes and the median set of median graphs. A natural question emerges: is there a linear-time algorithm computing the diameter and the radius for median graphs? We answer positively to this question for median graphs $G$ with constant dimension $d$, i.e. the dimension of the largest induced hypercube of $G$. We propose a combinatorial algorithm computing all eccentricities of median graphs with running time $O(2^{O(d…
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 grap…
Diameter and radius of an additive-multiplicative graph
该研究定义了正整数上由差1或素数商连接相邻顶点的图G，证明了其直径在6到7之间、半径在4到5之间，并在迪克森猜想下精确确定了直径和半径分别为6和4。
Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where $d_{ij}=d( v_i,v_j ) $. Since $D( G )$ is a non-negative real symmetric matrix, its eigenvalues can be arranged $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$, where eigenvalues $λ_1(G)$ and $λ_n(G)$ are called the distance spectral radius and the least distance eigenvalue of $G$, respectively. The {\it diameter} of graph $G$ is the farthest distance between all pa…
This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and quasi-radius. Moreover, maximal digraphs with finite values of radius and quasi-diameter are studied.
On the Steiner $k$-diameter and Steiner ($k,k^{\prime}$)-radius of trees
本文研究了树的Steiner k-直径和Steiner (k,k′)-半径，推广了Chartrand等人1989年的结果，并为k′=2和3时给出了Steiner k-直径的紧上界。
We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of Stevanovic, Zhang, and Alon and Sudakov.
A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root. The pebbling number of a graph $π(G)$ is the minimum number of pebbles needed so every distribution of $π(G)$ pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations…
This paper continues the results of "Domination Cover Pebbling: Graph Families." An almost sharp bound for the domination cover pebbling (DCP) number for graphs G with specified diameter has been computed. For graphs of diameter two, a bound for the ratio between the cover pebbling number of G and the DCP number of G has been computed. A variant of domination cover pebbling, called subversion DCP is introducted, and preliminary results are discussed.
The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and unicyclic graphs respectively.We also investigate the spectral radius of the square of unicyclic graphs with given girth or trees with fixed diameter.
Let $G$ and $H$ be graphs, and $G\boxtimes H$ the strong product of $G$ and $H$. We prove that for any connected graphs $G$ and $H$ there is a strongly connected orientation $D$ of $G\boxtimes H$ such that ${\rm diam}(D)\leq 2r+15$, where $r$ is the radius of $G\boxtimes H$.This improves the general bound ${\rm diam}(D)\leq 2r^2+2r$ for arbitrary graphs, proved by Chv${\rm \acute{a}}$tal and Thomassen.
Let $G$ be an irregular graph on $n$ vertices with maximum degree $Δ$ and diameter $D$. We show that Δ-λ_1>\frac{1}{nD} where $λ_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.
In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We show the connection between minimizing the total distance and maximizing the size under the same conditions. We also prove some asymptotically optimal bounds for the maximum total distance.
Matching Cut and Variants on Bipartite Graphs of Bounded Radius and Diameter
该研究探究匹配割及其多个变体问题在有界半径和直径二分图上的计算复杂度，拓展了已有结论，给出部分问题的复杂度二分法，解决了一个未决情形并得出首个难度结果。
Finding d-Cuts in Graphs of Bounded Diameter, Graphs of Bounded Radius and H-Free Graphs
本文系统研究特殊图类上的 d-Cut 问题，证明若干图类对 d≥2 可多项式求解，并给出 NP 困难性结果，形成有界直径、半径及无 H 图的复杂度二分图景。
We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show that the graph diameter for the graph reduced to a box of side $L$ scales like $(\log L)^{Δ+o(1)}$ where $Δ^{-1}:=\log_2(2d/s)$. In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance $L$. We also show that a ball of radius $r$ in th…
For any natural number n, the Hurwitz graph $\mathcal{H}(S_n )$ has vertices corresponding to reduced words of the cycle $(1 . . . n)$ in terms of transpositions in $S_n$, and has edges corresponding to local shifts. This graph is known to be connected, and some of its graph theoretic properties, such as its radius and upper bounds on its diameter, are known. We explore further the structure of the Hurwitz Graph, in particular its center and its connection to the geometric tree graph $G_n$. Finally, the extended Hurwitz Graph is defined and its properties such as the radius, characterization o…
We consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. We show that it is NP-complete to decide whether a graph has an orientation such that the sum of all the shortest paths lengths is at most an integer specified in the input. The proof is a short reduction from a result of Chvátal and Thomassen showing that it is NP-complete to decide whether a graph can be oriented so that its diameter is at most 2. In contrast, for each positive integer k, we describe a linear-time algorithm that decides fo…
The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of $rad(G)$ and $η(G)$, wher…
In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We also state an analog of the conjecture of Chen et al for digraphs and prove it for sufficiently large order.
When one studies geometric properties of graphs, local finiteness is a common implicit assumption, and that of transitivity a frequent explicit one. By compactness arguments, local finiteness guarantees several regularity properties. It is generally easy to find counterexamples to such regularity results when the assumption of local finiteness is dropped. The present work focuses on the following problem: determining whether these regularity properties still hold when local finiteness is replaced by an assumption of transitivity. After recalling the locally finite situation, we show that there…
We prove the nonexistence of lattice tilings of $\mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $n\geq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a prime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound.
Graph pebbling is a network model for transporting discrete resources that are consumed in transit. Deciding whether a given configuration on a particular graph can reach a specified target is ${\sf NP}$-complete, even for diameter two graphs, and deciding whether the pebbling number has a prescribed upper bound is $Π_2^{\sf P}$-complete. Recently we proved that the pebbling number of a split graph can be computed in polynomial time. This paper advances the program of finding other polynomial classes, moving away from the large tree width, small diameter case (such as split graphs) to small tr…
A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every pair of distinct vertices of $G$ is connected by a rainbow path. It is known that for every integer $k\geq 2$ deciding if a graph $G$ has $rc(G)\leq k$ is NP-Hard, and a graph $G$ with $rc(G)\leq k$ has diameter $diam(G)\leq k$. In foregoing papers, we showed that a bridgeless graph with diameter 2 has rainbo…
A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $\mathbb R^4$ and on a three-dimensional sphere. We prove an analogue of Vázsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $1/\sqrt 2$. We prove Schur's conjecture for diameter graphs in $\mathbb R^4.$ We also establish the maximum number of triangles a d…
For $0\le α\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_α(G)=αD(G)+(1-α)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $α$-spectral radius of $G$ is the largest eigenvalue of $A_α(G)$. We give upper bounds for $α$-spectral radius for unicyclic graphs $G$ with maximum degree $Δ\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $α$-spectral radius…
Dichotomies for Maximum Matching Cut: $H$-Freeness, Bounded Diameter, Bounded Radius
研究 Maximum Matching Cut 的计算复杂性，在有界直径、有界半径和无 $H$ 图上给出完整二分性，并据此证明 Disconnected Perfect Matching 在无 $(P_6+sP_2)$ 图上多项式可解。
Let $q(H)$ be the signless Laplacian spectral radius of a graph $H$. In this paper, we prove that \\1. Let $H$ be a proper subgraph of a $Δ$-regular graph $G$ with $n$ vertices and diameter $D$. Then $$2Δ- q(H)>\frac{1}{n(D-\frac{1}{4})}.$$ \\2. Let $H$ be a proper subgraph of a $k$-connected $Δ$-regular graph $G$ with $n$ vertices, where $k\geq 2$. Then $$2Δ-q(H)>\frac{2(k-1)^{2}}{2(n-Δ)(n-Δ+2k-4)+(n+1)(k-1)^{2}}.$$ Finally, we compare the two bounds. We obtain that when $k>2\sqrt{\frac{(n-Δ)(n+Δ-4)}{n(4D-3)-2}}+1$, the second bound is always better than the first. On the other hand, when $k<…
For a group $G,$ the enhanced power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x, y$ are adjacent if and only if there exists an element $w$ in $G$ such that both $x$ and $y$ are powers of $w.$ The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set $G \setminus S,$ where $S$ is the set of dominating vertices of the enhanced power graph. In this paper, we first characterize the dominating vertices of enhanced power graph of any finite nilpotent group. Thereafter, we classify all nilpotent groups $G$ such that the proper en…
Characterization and chromatic number of triangle-free graphs with diameter 2
本文修正了Shekarriz等人对直径为2的无三角形图的刻画中的小缺陷，并利用该刻画证明了关于这类图色数的若干结果。
Shrinking the Jung radius: Maximizing partial coverage of finite point sets
本文研究了有限平面点集在不同半径下的最大覆盖问题，证明了半径1/2和1/4时的精确最大值，并给出了相关函数的上下界。
The diameter of KPKVB random graphs
本文研究了KPKVB随机图模型中各连通分量直径的性质，证明其最大直径几乎肯定为O(log N)，并给出了紧致的上界。
In this paper we prove Schur's conjecture in $\mathbb R^d$, which states that any diameter graph $G$ in the Euclidean space $\mathbb R^d$ on $n$ vertices may have at most $n$ cliques of size $d$. We obtain an analogous statement for diameter graphs with unit edge length on a sphere $S^d_r$ of radius $r>1/\sqrt 2$. The proof rests on the following statement, conjectured by F. Morić and J. Pach: given two unit regular simplices $Δ_1,Δ_2$ on $d$ vertices in $\mathbb R^d$, either they share $d-2$ vertices, or there are vertices $v_1\in Δ_1,v_2\in Δ_2$ such that $\|v_1-v_2\|>1$. The same holds for …
In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and challenging open question. For graphs with diameter 2 we provide the first subexponential algorithm with complexity $2^{O(\sqrt{n\log n})}$, which is asymptotically the same as the currently best known time complexity for the graph isomorphism (GI) problem. Moreover, we prove that the graph isomorphism problem on 3-colorable graphs with diameter 2 is GI-complete…
For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free g…
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