Michael J. Dinneen, Paul R. Hafner
The results of computer searches for large graphs with given (small) degree and diameter are presented. The new graphs are Cayley graphs of semidirect products of cyclic groups and related groups. One fundamental use of our ``dense graphs'' is in the design of efficient communication network topologies.
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This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach used to establish the results for degree 6 and 7 has been extended successfully to degree 8 and 9. Candidate graphs are defined as functions of the diameter for both degree 8 and degree 9. They are proven to be extremal for small diameters. They establish new lower bounds for all greater diameters, and are conjectured to be extremal. The existence of the degre…
This paper considers the degree-diameter problem for undirected circulant graphs. For degrees 10 and 11 newly discovered families of circulant graphs of arbitrary diameter are presented which are largest known and are conjectured to be extremal. They are also the largest-known Abelian Cayley graphs of these degrees. For each such family the order of every graph in the family is defined by a quintic polynomial function of the diameter which is specific to the family. The elements of the generating set for each graph are similarly defined by a set of polynomials in the diameter. The existence of…
We give two lower bounds on the largest order of an arc-transitive graph of diameter two and a given degree.
For $n \in \mathbb{N}$ let $δ_n$ be the smallest value such that every graph of order $n$ and minimum degree at least $δ_n$ admits an orientation of diameter two. We show that $δ_n=\frac{n}{2} + Θ(\ln n)$.
In this paper we give graphs with the largest known order for a given degree $Δ$ and diameter $D$. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the $(Δ, D)$ table for graphs.
The degree/diameter problem is the problem of finding the largest possible number of vertices $n_{Δ,D}$ in a graph of given degree $Δ$ and diameter $D$. We consider the problem for the case of diameter $D=2$. William G Brown gave a lower bound of the order of $(Δ,2)$-graph. In this paper, we give a generalization of his construction and improve the lower bounds for the case of $Δ=306$ and $Δ=307$. One is $(306,2)$-graph with $88723$ vertices, the other is $(307,2)$-graph with $88724$ vertices.
We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for "stitching" together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum s…
The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $Δ$ and diameter $k$. For fixed $k$, the answer is $Θ(Δ^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $Θ(Δ^{k-1})$, and for graphs of bounded arboricity the answer is $Θ(Δ^{\floor{k/2}})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given t…
The Maximum Degree and Diameter Bounded Subgraph Problem (MaxDDBS) asks: given a host graph G, a bound on maximum degree Δ, and a diameter D, what is the largest subgraph of the host graph with degree bounded by Δand diameter bounded by D? In this paper, we investigate this problem when the host graph is the k-dimensional mesh. We provide lower bounds for the size of the largest subgraph of the mesh satisfying MaxDDBS for all k and Δ> 3 that agree with the known upper bounds up to the first two terms, and show that for Δ= 3, the lower bounds are at least the same order of growth as the upper b…
Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3
本文证明了直径为2且最小度数至少为3的图中必存在长度为4或8的环，从而支持了埃尔多斯-加法斯猜想。
Table of large graphs with given degree and diameter
更新了具有给定度数和直径的大型无向图的表格，包含自2013年以来的新研究成果。
This is the full proof of Theorem 3 on the existence of the largest known degree 8 circulant graph for all diameters stated in the paper "The degree-diameter problem for circulant graphs of degree 8 and 9" by the author. To avoid the paper being unduly long the exceptions for only one case were included in the statement of the proof. In the statement of the proof presented in this paper the exceptions for all cases are included in full. This proof closely follows the approach taken by Dougherty and Faber for the existence of the largest known degree 6 circulant graph for all diameters.
It is known that the number of vertices of a graph of diameter two cannot exceed $d^2+1$. In this contribution we give a new lower bound for orders of Cayley graphs of diameter two in the form $C(d,2)>0.684d^2$ valid for all degrees $d\geq 360756$. The result is a significant improvement of currently known results on the orders of Cayley graphs of diameter two.
The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdös and Rényi. In this paper, we generalize the above problem for Steiner $k$-diameter, and study the problem of determining the minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$ a…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we restrict attention to Cayley graphs, and study the asymptotics by fixing a small diameter and constructing families of graphs of large order for all values of the maximum degree. Much of the literature in this direction is focused on the diameter two …
The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight.
The Steiner diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdös and Rényi. Recently, Mao considered the problem of determining the minimum size of a graph of order $n$ whose Steiner $k$-diameter is at most $d$ and whose maximum is at most $\ell$, where $3\leq k\leq n$, and…
We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.
One of the prominent areas of research in graph theory is the degree-diameter problem, in which we seek to determine how many vertices a graph may have when constrained to a given degree and diameter. Different variants of this problem are obtained by considering further restrictions on the graph, such as whether it is directed, undirected, mixed, vertex-transitive etc. The currently best known extremal constructions are the Gómez graphs, both when considered as directed and undirected. As the Gómez graphs are similar to the Faber-Moore-Chen graphs, we give a natural generalisation of their de…
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.
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by using a generalization to $\mathbb{Z}^n$ of the concept of congruence in $\mathbb{Z}$. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.
Let $N_2DL(v)$ denote the set of degrees of vertices at distance 2 from $v$. The $2$-neighborhood degree list of a graph is a listing of $N_2DL(v)$ for every vertex $v$. A degree restricted $2$-switch on edges $v_1v_2$ and $w_1w_2$, where $deg(v_1)=deg(w_1)$ and $deg(v_2)=deg(w_2)$, is the replacement of a pair of edges $v_1v_2$ and $w_1w_2$ by the edges $v_1w_2$ and $v_2w_1$ given that $v_1w_2$ and $v_2w_1$ did not appear in the graph originally. Let $G$ and $H$ be two graphs of diameter 2 on the same vertex set. We prove that $G$ and $H$ have the same $2$-neighborhood degree list if and only…
In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.
We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree $d$. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically $d^2/2$.
In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4 and 1/2 respectively. Recent papers have presented constructions which increase the coefficient of the lower bound to be at or just below 3/8, but only for sparse sets of degree d related to primes of specific congruence classes. By applying results from number theory these constructions can be extended to be valid for every degree above some threshold, esta…
The conditional diameter of a connected graph $Γ=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(Γ_1, Γ_2)$ of subgraphs of $Γ$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em diameter} measures the maximum distance among subgraphs satisfying ${\cal P}$. That is, \[ D_{\cal P}(Γ):=\max_{Γ_1, Γ_2\subset Γ} \{\partial(Γ_1, Γ_2): Γ_1, Γ_2 \quad {\rm satisfy }\quad {\cal P}\}. \] In this paper we consider the conditional diameter in which ${\cal P}$ requires that $δ(u)\ge α$ for all $ u\in V(Γ_1)$, $δ(v)\ge β$ for all $v\in V(Γ_2)$, $| V(Γ_1)| \ge s$ and $| V(Γ_…
Brightwell, van den Heuvel and Stougie proved that the diameter of an $m \times n$ transportation polytope is at most $8(m+n-2)$, a factor of eight away from the Hirsch Conjecture. This bound was improved to $3(m+n-1)$ by Hurkens. We investigate diameters for certain classes of transportation polytopes. Note: After the completion of this note, we discovered that the class of transportation polytopes studied in this note was already considered in Michel L. Balinski. On two special classes of transportation polytopes. Math. Programming Stud., 1:43-58, 1974. Michel L. Balinski and Fred J. Rispoli…
We address the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs), both directed and undirected. The problem turns out to be closely related to the problem of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. In particular, we find the largest Abelian graphs with 2 generators (dimensions) and a given diameter. (The results for 2 generators are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with 3 generators and…
A graph is called diameter-$k$-critical if its diameter is $k$, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-$k$-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Häggkvist (that in every diameter-2-critical graph, the average edge-degree is at most the number of vertices), which promised to completely solve the extremal problem for diameter-2-critical graphs. On the other hand, w…
A mixed graph is said to be dense if its order is close to the Moore bound and it is optimal if there is not a mixed graph with the same parameters and bigger order. We present a construction that provides dense mixed graphs of undirected degree $q$, directed degree $\frac{q-1}{2}$ and order $2q^2$, for $q$ being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to $\frac{9q^2-4q+3}{4}$ the defect of these mixed graphs is $({\frac{q-2}{2}})^2-\frac{1}{4}$. In particular we obtain a known mixed Moore graph of order $18$, undirected degree $3$ and directe…
On the oriented diameter of graphs with given minimum degree
本文研究给定最小度图的有向直径上界问题，证明使有向直径满足 c·3n/(δ+1)+O(1) 的最小常数 c 为 1，解决了 Bau 和 Dankelmann 提出的最优常数问题。
We analyze the following version of the deterministic \hats game. We have a graph $G$, and a sage resides at each vertex of $G$. When the game starts, an adversary puts on the head of each sage a hat of a color arbitrarily chosen from a set of $k$ possible colors. Each sage can see the hat colors of his neighbors but not his own hat color. All of sages are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The strategy is winning if it guarantees at least one correct ind…
Diameters and mixing times for giant components of random graphs with given degrees
研究随机图中给定度数序列的大连通分量的直径和混合时间，证明其唯一性并给出界限，同时分析其他分量的大小和直径。
The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the \emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic mixed graph with given minimum undirected and directed degrees; this is a generalisation of the classical degree/girth problem. In this paper we present new bounds on the order of mixed graphs with given diameter or geodetic girth and exhibit new examples of directed and mixed geodetic cages. In particular, we show that any $k$-geodetic mixed graph with excess one …
The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree $Δ$ and the diameter $D$, was introduced in \cite{maxddbs}, as a generalization of the Degree-Diameter Problem. A case of special interest is when the host graph is a common parallel architecture. Here we discuss the case when the host graph is a $k$-dimensional mesh. We provide some general bounds for the order of the largest subgraph in arbitrary dimension $k$, and for the particular cases of $k=3, Δ= 4$ and $k=2, Δ= 3$, we give constructions that result in sha…
Erdős, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $δ\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}δ + O(1)$ for every $r\ge 2$, if $δ$ is a multiple of $(r-1)(3r+2)$. For every $r>1$ and $δ\ge 2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $δ$ and diameter $\frac{(6r-5)n}{(2r-1)δ+2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $δ>2(r-1)(3r+2)(2r-3)$. The rest of the paper proves positive results under a stronger hypot…
Weak diameter choosability of graphs with an excluded minor
本文扩展了Dvořák和Norin的结果，证明了对于任意图H，H-无子式图具有有界弱直径的最优3-可选性，并在H为平面图时证明了更强的预着色扩展结果及指数级着色存在性，同时指出该列表着色结果无法扩展至奇无子式图。
The degree-diameter problem for plane graphs with pentagonal faces
本文研究直径为3的五边形剖分平面图的度-直径问题，证明当最大度 $Δ\geq8$ 时顶点数满足 $n\leq3Δ-1$，且该界在 $Δ$ 为奇数时紧。
This paper considers the degree-diameter problem for extremal and largest known undirected circulant graphs of degree 2 to 9 of arbitrary diameter. As these graphs are vertex transitive it is possible to define their distance partition. The number of vertices in each level of the distance partition is shown to be related to an established upper bound for the order of Abelian Cayley graphs. Furthermore these graphs are all found to have odd girth which is maximal for their diameter. Therefore the type of each vertex in a level may be well-defined by the number of adjacent vertices in the preced…
We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree $Δ$ with more than $1.5 Δ^t$ edges, its line graph must have diameter larger than $t$. In the case where the graph contains no cycle of length $2t+1$, we can improve the bound on the number of edges to one that is exact for $t\in\{1,2,3,4,6\}$. In the case $Δ=3$ and $t=3$, we obtain an exact bound. Our results also have implications for the related problem of bounding the dis…
Non-Isomorphic Optimal Cayley Graphs
本文构造了七十多个与已知最优Cayley图非同构的度直径最优Cayley图，通过全面比较图不变量，揭示了最优Cayley图的非唯一性，并扩充了已知最优图目录。
Almost Moore mixed graphs\/} appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter $2$. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal constructions for other diameters.
Cayley graph techniques are introduced for the problem of constructing networks having the maximum possible number of nodes, among networks that satisfy prescribed bounds on the parameters maximum node degree and broadcast diameter. The broadcast diameter of a network is the maximum time required for a message originating at a node of the network to be relayed to all other nodes, under the restriction that in a single time step any node can communicate with only one neighboring node. For many parameter values these algebraic methods yield the largest known constructions, improving on previous …
We address the problem of constructing large undirected circulant networks with given degree and diameter. First we discuss the theoretical upper bounds and their asymptotics, and then we describe and implement a computer-based method to find large circulant graphs with given parameters. For several combinations of degree and diameter, our algorithm produces the largest known circulant graphs. We summarize our findings in a table, up to degree 15 and diameter 10, and we perform a statistical analysis of this table, which can be useful for evaluating the performance of our methods, as well as o…
Largest planar graphs of diameter $3$ and fixed maximum degree -- connection with fractional matchings
针对直径3、最大度固定的平面图，本文借助分数最大匹配证明最大顶点数上界为9/2Δ+c，并刻画极值图。
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 degree of each edge over all edges. In our work, we explore the multiplicative Zagreb indices of bipartite graphs of order $n$ with diameter $d$, and sharp upper bounds are obtained for these indices of graphs in $\mathcal{B}(n,d)$, where $\mathcal{B}(n, d)$ is the set of all $n$-vertex bipartite graphs with the diameter $d$. In addition, we explore the relationship between the maximal multiplicative Zagreb indices of graph…
The harmonic index of a graph $G$, is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ is the degree of the vertex $u$ in $G$. In this paper we find the minimum harmonic index of bicyclic graph of order $n$ and diameter $d$. We also characterized all bicyclic graphs reaching the minimum bound.
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) Δ(T)$. This implies a polynomial-time constant-factor appro…
We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this context, the bipartite Moore bound $\M^b(d,D)$ represents a general upper bound for $\N^b(d,D)$. Bipartite graphs of order $\M^b(d,D)$ are very rare, and determining $\N^b(d,D)$ still remains an open problem for most $(d,D)$ pairs. This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on bipartite $(d,D,-4)$-graphs (that is, bipartite gr…
On the Total Regularity of Almost Mixed Moore Graphs
本文解决了几乎混合摩尔图的全正则性开放问题，证明了3个直径至少为3的已知几乎混合摩尔图是仅有的此类混合图，相关成果可应用于网络设计领域。
Connectivity and Diameter of Friends and Strangers Graphs Using Minimum Degree
针对友敌图的连通性与直径问题，基于最小度条件，证明当图满足一定最小度要求时友敌图连通且直径达到最优界，改进了相关研究成果并解决了部分问题。
We study a family of graphs with diameter two and asymptotically optimal order for their maximum degree, obtained from perfect difference sets. We show that for all known examples of perfect difference sets, the graph we obtain is isomorphic to one of the Brown graphs, a well-known family of graphs in the degree-diameter problem.
The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of ve…
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs $MCK(d,\ell)$ and it is derived from the Kautz digraphs $K(d,\ell)$. %It is not common to find non-regular digraphs with minimal diameter given their number of vertices and out-degree. It is well-known that the Kautz digraphs $K(d,\ell)$ h…
Height functions on the $m \times n$ Miura-ori flip graph: degree sequence and diameter
研究 $m \ imes n$ Miura-ori 翻转图中顶点度序列和直径，通过将每个赋值映射为网格上的整数高度函数，用显式多项式计数各度数的顶点，并给出直径的闭式下界和匹配上界。
Diameter two orientability of mixed graphs
本文确定了混合图（同时含无向边和有向边）实现直径为二的定向所需的最小度阈值，将无向图的已知结果推广至混合图，为定向优化提供更普适的充分条件。
A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give a simple criterion based on the degree sequence that says whether a degree sequence will have a pseudoline arrangement graph as one of its realizations. In the second part, we study the eccentricities of vertices in such graphs. We observe that the diameter (maximum eccentricity of a vertex in the graph) of any pseudoline arrangement graph on $n$ pseudolines…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree $r$ and directed out-degree $z$, a straightforward counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of the graph. Apart from the case $r=1$, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there ar…
The Randić index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randić index $R(G)$ with relations to the diameter $D(G)$ and the average distance $μ(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $δ(G)$, if $δ(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $δ(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq μ(G)$. Further…
In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $δ$ there is an orientation of diameter at most $11\frac{n}{δ+1}+9$. In 2016, Surmacs reduced this bound to $7\frac{n}{δ+1}.$ In this paper, we consider the girth of a graph $g$ and show that for any $\varepsilon>0$ there is a bound of the form $(2g+\varepsilon)\frac{n}{h(δ,g)}+O(1)$, where $h(δ,g)$ is a polynomial. Letting $g=3$ and $\varepsilon<1$ gives an inprovement on the result by Surmacs.
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