Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a ...

• Characterize the extremal graphs having the maximum spectral radius in some given class of graphs.
• Establish a sharp lower bound on the generalized 4-independence number of a tree with fixed order.
• Describe the structure of ...

A network for the transportation of supplies can be described as a rooted tree with a weight of a degree of congestion for each edge. We take the sum of root-leaf distance (SRD) on a rooted tree as the whole degree of congestion of the tree. Hence,...${}_{}$$$${}^{\mathrm{}}$$$${}_{}$

For an oriented tree, we compute several graph invariants, including the minimal norm of the generalized inverse and the norm of the Moore–Penrose inverse of its incidence matrix. We present equivalent characterizations of these invariants in ...

In this paper, we delve into the investigation of locating broadcast 2-centers of a tree T under the postal model. The problem asks to deploy two broadcast centers so that the maximum communication time from the centers to their corresponding ...

A mobile agent, which is an autonomous device navigating in a graph, has to explore a given graph by visiting all of its nodes. We adopt the (arguably) weakest possible model of such a device: a deterministic memoryless automaton (DMA), i.e., a ...

Given a connected graph G = ( V , E ) and a vertex subset S ⊂ V, the Steiner distance d G ( S ) of S is the size of a minimum spanning tree of S in G. If G has order n and k is an integer such that 2 ≤ k ≤ n, the Steiner k-eccentricity of a ...

In light of the successful investigation of the adjacency matrix, a significant amount of its modification is observed employing numerous topological indices. The matrix corresponding to the well-known first Zagreb index is one of them. The ...${}_{{}_{}}{}_{{}_{}}$${}_{}$${}_{}$${}_{{}_{}}$${}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}$

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric functions. ...

Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph G are contaminated initially, before the process ...

We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of ...

The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path in G contains three or more vertices of S. The general ...

We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of caterpillars, to Laplacian limit points. We show that this ...

• We investigate the Hoffman concept of limit points for the Laplacian matrix of a graph. The problem is to determine which real numbers are limit points of the spectral radius of these matrices.
• We make progress towards the proof that ...

A signed graph G ˙ is defined to be harmonic if its net-degree vector d ± ( G ˙ ) is a zero vector or an eigenvector of adjacency matrix A ( G ˙ ) corresponding to an eigenvalue λ. In this paper, we demonstrate the existence of λ-harmonic signed ...

• The existence of harmonic signed tree with arbitrary diameter is proved.
• The nonexistence of finite 0-harmonic signed trees is proved.
• All harmonic signed trees with diameter less than 6 are determined.
• Some general properties ...

A given graph H is called realizable by a graph G if $G[N(v)]\cong H$ for every vertex v of G. The Trahtenbrot-Zykov problem says that which graphs are realizable? We consider a problem somewhat opposite in a more general setting. Let $\mathcal{F}$ be a family of ...$\mathcal{}$${\mathcal{}}_{}$$\mathrm{}$$\mathcal{}{}_{}\mathrm{}$$\mathcal{}_{\mathrm{}}^{}\mathrm{}$$\stackrel{}{}$${\mathcal{}}_{}$$$$_{\mathrm{}}^{}$$$$_{\mathrm{}}^{}$$$$$$\mathcal{}$$$$\mathcal{}$$$$$

Let G be a simple connected graph and A ( G ) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A ( G ). A nonsingular graph G is said to have the reciprocal eigenvalue property if the ...

A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph K n ∖ C k, obtained from the complete graph K n with n vertices by deleting all edges of a cycle C k with k ...

A general exponential vertex-degree-based topological index (exponential VDB index for short) T e f of a graph G is the summation of e f ( d ( u ) , d ( v ) ), where f ( x , y ) is a symmetric real function with x ≥ 1 and y ≥ 1, and the summation ...

• A exponential vertex-degree-based topological index is a general topological index.
• It is very important to characterize the extremal graphs for some topological indices.
• The path minimizes some exponential vertex-degree-based ...

For stacked simplicial complexes, (special subclasses of such are: trees, triangulations of polygons, stacked polytopes with their triangulations), we give an explicit bijection between partitions of facets (for trees: edges), and partitions of ...$$$\mathrm{}$

We study three combinatorial models for the lower-triangular matrix with entries t n , k = ( n k ) n n − k: two involving rooted trees on the vertex set [ n + 1 ], and one involving partial functional digraphs on the vertex set [ n ]. We show ...

A graph G is said to be determined by its generalized spectrum (DGS for short), if whenever H is a graph such that H and G are cospectral with cospecral complements then H is isomorphic to G. Let G be an n-vertex graph with adjacency matrix A and ...