Unicyclic and bicyclic graphs having minimum degree distance

In this paper characterizations of connected unicyclic and bicyclic graphs in terms of the degree sequence, as well as the graphs in these classes minimal with respect to the degree distance are given.     

Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P₄, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K₁+(K_1,k+1∪(n−k−3)K₁).     

On the degree distance of a graph

If G is a connected graph with vertex set V, then the degree distance of G, D^′(G), is defined as , where degw is the degree of vertex w, and d(u,v) denotes the distance between u and v. We prove the asymptotically sharp upper bound for graphs of order n and diameter d. As a corollary we obtain the bound for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].     

Proximity and remoteness in triangle-free and C4-free graphs in terms of order and minimum degree

Let G be a finite, connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness $\rho (G)$ and the proximity $\pi (G)$ of G are the maximum and the minimum of the average distances of the vertices of G. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of $C_4$-free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible.     

Large bipartite Cayley graphs of given degree and diameter

Let BC_d,k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k. We show that BC_d,k≥2(k−1)((d−4)/3)^k−1 for any d≥6 and any even k≥4, and BC_d,k≥(k−1)((d−2)/3)^k−1 for d≥6 and k≥7 such that k≡3 (mod 4).     

Hamiltonian graphs of given order and minimum algebraic connectivity

In this paper, we study the algebraic connectivity of a Hamiltonian graph, and determine all Hamiltonian graphs whose algebraic connectivity attain the minimum among all Hamiltonian graphs on n vertices.     

Four new sums of graphs and their Wiener indices

The Wiener index is the sum of distances between all vertex pairs in a connected graph. This notion was motivated by various mathematical properties and chemical applications. In this paper we introduce four new operations on graphs and study the Wiener indices of the resulting graphs.     

Diameter of orientations of graphs with given minimum degree

We show that every bridgeless graph of order $n$ and minimum degree $\delta$ has a strongly connected orientation of diameter at most $\frac{11}{\delta +1}n+9$, and such an orientation can be found in polynomial time.     

The signless Laplacian spectral radius of graphs with given degree sequences

In this paper, we investigate the properties of the largest signless Laplacian spectral radius in the set of all simple connected graphs with a given degree sequence. These results are used to characterize the unicyclic graphs that have the largest signless Laplacian spectral radius for a given unicyclic graphic degree sequence. Moreover, all extremal unicyclic graphs having the largest signless Laplacian spectral radius are obtained in the sets of all unicyclic graphs of order n with a specified number of leaves or maximum degree or independence number or matching number.     

On the minimum order of graphs with given semigroup

Denote by M(n) the smallest positive integer such that for every n-element monoid M there is a graph G with at most M(n) vertices such that End(G) is isomorphic to M. It is proved that $\text{2}\text{(1 + o(1))n}\text{log}{}_2\text{n}\text{≤M(n)≤n ·}\text{2n}\text{+ O(n)}$. Moreover, for almost all n-element monoids M there is a graph G with at most 12 · n · log₂n + n vertices such that End(G) is isomorphic to M.     

Regular separable graphs of minimum order with given diameter

A (d, c, v)-graph G is one which is regular of degree v and has diameter d and connectivity c. G is said to be minimum if it is of minimum order, i.e. has the minimumnumber of points; G is separable if c=1.In this paper, the minimum order of a (d, 1, v)-graph is determined and the construction of all minimum (d, 1, v)-graphs is described.     

A note on minimum degree conditions for supereulerian graphs

A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut SE(G) with |S|3 satisfies the property that each component of G−S has order at least (n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4: If G is a 2-edge-connected graph of order n with δ(G)4 such that for every edge xyE(G) , we have max{d(x),d(y)}(n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ(G)4 cannot be relaxed.     

The number of cycle lengths in graphs of given minimum degree and girth

This paper determines lower bounds on the number of different cycle lengths in a graph of given minimum degree k and girth g. The most general result gives a lower bound of ck^g/8.     

On the excess of vertex-transitive graphs of given degree and girth

We consider a restriction of the well-known Cage Problem to the class of vertex-transitive graphs, and consider the problem of finding the smallest vertex-transitive $k$-regular graphs of girth $g$. Counting cycles to obtain necessary arithmetic conditions on the parameters $(k,g)$, we extend previous results of Biggs, and prove that, for any given excess $e$ and any given degree $k\ge 4$, the asymptotic density of the set of girths $g$ for which there exists a vertex-transitive $(k,g)$-cage with excess not exceeding $e$ is 0.