Related papers: Complex unit gain bicyclic graphs with rank 2, 3 o…

Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…

A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of…

F-index of a graph is the sum of the cube of the degrees of the vertices. In this paper, we investigate the F-indices of unicyclic graphs by introducing some transformation, and characterize the unicyclic graphs with the first five largest…

The general spectral radius of a graph $G$, denoted by $\Theta(G,\alpha)$, is the maximal eigenvalue of $M_{\alpha}(G)=A(G)+\alpha D(G)$ $(\alpha\geq 0)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex…

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a…

The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph…

For a finite group $G$, the vertices of the prime graph $\Gamma(G)$ are the primes that divide $|G|$, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in $G$. Prime graphs of solvable…

Let $G$ be a finite group and $S$ be a subset of $G$. The bi-Cayley graph $\mathrm{BCay}(G,S)$ is the graph with vertex set $G\times \{0,1\}$ and edge set $\{\{(x,0),(sx,1)\}\mid x\in G,s\in S\}$. A bi-Cayley graph $\mathrm{BCay}(G,S)$ is…

The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue…

In this paper, we study dual quaternion and dual complex unit gain graphs and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of…

A graph $G$ on $n$ vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to $n$. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

The power graph of a group $G$ is a simple and undirected graph with vertex set $G$ and two distinct vertices are adjacent if one is a power of the other. In this article, we characterize (non-cyclic) finite groups of prime exponent and…

Let $R$ be a finite ring with identity. The unit graph (unitary Cayley graph) of $R$ is the graph with vertex set $R$, where two distinct vertices $x$ and $y$ are adjacent exactly whenever $x+y$ is a unit in $R$ ($x-y$ is a unit in $R$).…

We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group.

Bilevel graph structure learning is widely understood to improve graph neural networks by jointly optimizing model parameters and a learned graph structure, with the resulting performance gain attributed to the rewired adjacency. We find…

Strongly regular graphs are regular graphs with a constant number of common neighbours between adjacent vertices, and a constant number of common neighbours between non-adjacent vertices. These graphs have been of great interest over the…

Let $G$ be a graph with nonnegative integer weights. A {\it unit acquisition move} transfers one unit of weight from a vertex to a neighbor that has at least as much weight. The {\it unit acquisition number} of a graph $G$, denoted…

The degree sequence of a graph is the sequence of the degrees of its vertices. If $\pi$ is a degree sequence of a graph $G$, then $G$ is a realization of $\pi$ and $G$ realizes $\pi$. Determining when a sequence of positive integers is…