The energy \(E(G)\) of a graph \(G\) is the sum of the absolute values of the eigenvalues of \(G\). Two graphs \(G_1\) and \(G_2\) are said to be equienergetic if \(E(G_1) = E(G_2)\). In this paper, we outline various classes of equienergetic graphs. These results enable the construction of pairs of noncospectral equienergetic graphs of the same order and of the same size.

In this paper, fuzzy finite state automaton with unique membership transition on an input symbol (uffsa) is defined. It is proved and illustrated that for a given fuzzy finite state automaton (ffsa), there exists an equivalent uffsa. Some closure properties of fuzzy regular languages are studied.

A \((G,H)\)-multifactorization of \(\lambda K_m\) is a partition of the edge set of \(\lambda K_m\) into \(G\)-factors and \(H\)-factors with at least one \(G\)-factor and one \(H\)-factor. Atif Abueida and Theresa O’Neil have conjectured that for any integer \(n \geq 3\) and \(m \geq n\), there is a \((G_n, H_n)\)-multidecomposition of \(\lambda K_m\) where \(G_n = K_{1,n-1}\) and \(H_n = C_n\). In this paper, it is shown that the above conjecture is true for \(m=n\) when

1. \(G_m = K_{1,m-1}; H_m = C_m\),
2. \(G_m = H_{1,m-1}; H_m = P_m\), and
3. \(G_m = P_m; H_m = C_m\).

For a path \( P_n \) of order \( n \) and for any odd integer \( k \), \( 1 \leq k \leq n – 3 \), Chartrand et al. have given an upper bound for the radio \( k \)-chromatic number of \( P_n \) as \( \frac{k^2+2k+1}{2} \). Here we improve this bound for \( \frac{n-4}{2} \leq k < \frac{2n-5}{3} \) and \( \frac{2n-5}{3} \leq k \leq n-3 \). They are \( \frac{k^2+k+4}{2} \) and \( \frac{k^2+k+2}{2} \), respectively. Also, we improve the lower bound of Kchikech et al. from \( \frac{k^2+3}{2} \) to \( \frac{k^2+5}{2} \) for odd integer \( k \), \( 3 \leq k \leq n-3 \).

In this paper, we obtain a necessary condition for the Skolem gracefulness of the disjoint union of \( k \) signed stars \( K_{1,r_i}, 1 \leq i \leq k \), which we call a \( k \)-signed star \( St(r_1,r_2,\ldots,r_k) \). We also present results on the Skolem gracefulness of the 2-signed star \( St(r_1,r_2) \).

In this paper, a definition of a variation of the standard notion of the line signed graph of a given signed graph is recalled from [14] and some fundamental results linking it to the notions of jump signed graphs [6] and adjacency signed graphs [21], especially with regard to their states of balance, consistency, and compatibility are obtained.

In this paper, we discuss how the addition of a new edge changes the irregularity strength in \( K_{m,m} \) and \( tC_4 \).

The goal of this article is to provide an overview of all the results currently known regarding the connectedness of path graphs. The proofs we present are only those that illustrate the different techniques employed in obtaining the results.

This is an expository paper addressed to readers with a small degree of familiarity with the field of graph theory and its techniques.

This paper presents some new results on permissible degree sets in polygon visibility graphs (PVGs). If the PVG has \( n \) vertices, we say it is an \( n \)-PVG. We also show some canonical construction techniques for PVGs with given degree sets.

In this paper, we present several graph theory related software systems that we have developed. These systems have been used in learning and research. The systems feature drawing and manipulation of graphs as well as execution of graph algorithms. The systems are: JGraph, a Java-based system for creating graphs and running graph algorithms; Colossus, a visibility graph system; Manohar, a system for computing graceful labelings of graphs (with special emphasis on trees); Graph Algorithm Constructor, which allows the creation of graph algorithms by drawing flow diagrams instead of writing source code. We also describe some examples in which the empirical data generated from these systems have allowed us to discover fundamental properties of graphs.