We prove that if two graphs of girth at least 6 have isomorphic squares, then the graphs themselves are isomorphic. This is the best possible extension of the results of Ross and Harary on trees and the results of Farzad et al. on graphs of girth at least 7. We also make a remark on reconstruction of graphs from their higher powers.

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks w...

We study 0-1 laws for extensions of rst-order logic by Lindstrr om quantiiers. We state suucient conditions on a quantiier Q expressing a graph property, for the logic FOQ] { the extension of rst-order logic by means of the quantiier Q { to have a 0-1 law. We use these conditions to show, in particular, that FORig], where Rig is the quantiier expressing rigidity, has a 0-1 law. We also show tha...

In a given graph G = (V , E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring ofG if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number. In this note, we study the chromatic n...

Buckley and Harary introduced several graphical invariants related to convexity theory, such as the geodetic number of a graph. These invariants have been the subject of much study and their determination has been shown to be NP -hard. We use the probabilistic method developed by Erdös to determine the asymptotic behavior of the geodetic number of random graphs with fixed edge probability. As a...

A map from E (G) to {1, 2, 3, ..., t} is called a t-linear coloring if (V (G), φ(α)) is a linear forest for 1 ≤ α ≤ t. The linear arboricity la (G) of a graph G defined by Harary [9] is the minimum number t for which G has a t-linear coloring. Let G be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove that if G contains no 4-cycles and intersecting triangles...

Let S be a set of transpositions such that the girth of the transposition graph of S is at least 5. It is shown that the automorphism group of the Cayley graph of the permutation group H generated by S is the semidirect product R(H) ⋊ Aut(H,S), where R(H) is the right regular representation of H and Aut(H,S) is the set of automorphisms of H that fixes S setwise. Furthermore, if the connected co...

A graph is said to be a sum graph if there exists a set S of positive integers as its node set, with two nodes adjacent whenever their sum is in S. An integral sum graph is defined just as the sum graph, the difference being that S is a subset of 2~ instead of N*. The sum number of a given graph G is defined as the smallest number of isolated nodes which when added to G result in a sum graph. T...

In 1857, the Irish mathematician Sir William Hamilton(1805-1865) invented a game of travelling around the edges of a graph from vertex to vertex. Hamilton described the game, in a letter to his friend Graves, as a mathematical game on the dodecahedron in which one person sticks five pins in any five consecutive vertices and the other is required to complete the path to form a spanning cycle. In...

Harary’s edge reconstruction conjecture states that a graph G=(V; E) with at least four edges is uniquely determined by the multiset of its edge-deleted subgraphs, i.e. the graphs of the form G − e for e∈E. It is well-known that this multiset uniquely determines the degree sequence of a graph with at least four edges. In this note we generalize this result by showing that the degree sequence of...