The graph’s irregularity is the sum of the absolute values of the differences of degrees of pairs of adjacent vertices in the graph. We provide various upper bounds for the irregularity of a graph, especially for \(K_{r+1}\)-free graphs, where \(K_{r+1}\) is a complete graph on \(r+1\) vertices, and trees and unicyclic graphs of given number of pendant vertices.

Let \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) be the \(n\)-dimensional vector (resp. affine) space over the finite field \(\mathbb{F}_q\). For \(1 \leq i \leq i+s \leq n-1\) (resp. \(0 \leq i \leq i+s \leq n-1\)), let \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) denote the set of all subspaces (resp. flats) in \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) with dimensions between \(i\) and \(i+s\) including \(\mathbb{F}_q^(n)\) and \(\{0\}\) (resp. \(\emptyset\)). By ordering \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) by ordinary inclusion or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity.

In this paper, we give some relations involving the usual Fibonacci and generalized order-\(k\) Pell numbers. These relations show that the generalized order-\(k\) Pell numbers can be expressed as the summation of the usual Fibonacci numbers. We find families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, \(F_{2i-1}\), and their sums. Also, extending these matrix representations, we find families of super-diagonal matrices such that the permanents of these matrices are the generalized order-\(k\) Pell numbers and their sums.

Let \(G\) be a finite group and \(S\) be a subset (possibly containing the identity element) of \(G\). We define the Bi-Cayley graph \(X = BC(G, S)\) to be the bipartite graph with vertices \(G \times \{0, 1\}\) and edges \(\{(g, 0), (sg, 1) : g \in G, s \in S\}\). In this paper, we show that if \(X = BC(G, S)\) is connected, then \(\kappa(X) = \delta(X)\).

Some new characterizations for harmonic Bergman space on the unit ball \({B}\) in \(\mathbb{R}^n\) are given in this paper. They can be described as derivative-free characterizations.

The planar Ramsey number \(PR(H_1, H_2)\) is the smallest integer \(n\) such that any planar graph on \(n\) vertices contains a copy of \(H_1\) or its complement contains a copy of \(H_2\). It is known that the Ramsey number \(R(K_4 – e, K_k – e)\) for \(k \leq 6\). In this paper, we prove that \(PR(K_4 – e, K_6 – e) = 16\) and show the lower bounds on \(PR(K_4 – e, K_k – e)\).

Let \(K_v\) be a complete graph with \(v\) vertices, and \(G = (V(G), E(G))\) be a finite simple graph. A \(G\)-design \(G-GD_\lambda(v)\) is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). In this paper, the existence of graph designs \(G-GD_\lambda(v)\), \(\lambda > 1\), for eight graphs \(G\) with six vertices and eight edges is completely solved.

A \({weighted \;graph}\) is one in which every edge \(e\) is assigned a nonnegative number \(w(e)\), called the \({weight}\) of \(e\). The \({weight\; of \;a \;cycle}\) is defined as the sum of the weights of its edges. The \({weighted \;degree}\) of a vertex is the sum of the weights of the edges incident with it. In this paper, motivated by a recent result of Fujisawa, we prove that a \(2\)-connected weighted graph \(G\) contains either a Hamilton cycle or a cycle of weight at least \(2m/3\) if it satisfies the following conditions:
\((1)\) The weighted degree sum of every three pairwise nonadjacent vertices is at least \(m\);\((2)\)In each induced claw and each induced modified claw of \(G\), all edges have the same weight.This extends a theorem of Zhang, Broersma and Li.

The \({restricted edge-connectivity}\) of a graph is an important parameter to measure fault-tolerance of interconnection networks. This paper determines that the restricted edge-connectivity of the de Bruijn digraph \(B(d,n)\) is equal to \(2d – 2\) for \(d \geq 2\) and \(n \geq 2\) except \(B(2,2)\). As consequences, the super edge-connectedness of \(B(d,n)\) is obtained immediately.

An edge coloring of a graph is called \({square-free}\) if the sequence of colors on certain walks is not a square, that is not of the form \(x_1, \ldots, x_m, x_{1}, \ldots, x_m\) for any \(m \in \mathbb{N}\). Recently, various classes of walks have been suggested to be considered in the above definition. We construct graphs, for which the minimum number of colors needed for a square-free coloring is different if the considered set of walks vary, solving a problem posed by Brešar and Klavžar. We also prove the following: if an edge coloring of \(G\) is not square-free (even in the most general sense), then the length of the shortest square walk is at most \(8|E(G)|^2\). Hence, the necessary number of colors for a square-free coloring is algorithmically computable.