The triangle graph of a graph G, denoted by T (G), is the graph whose vertices represent the triangles (K 3 subgraphs) of G, and two vertices of T (G) are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of C n -free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs G for which T (G) is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.

Abstract. We introduce three types of extension of Gallai graph operator in to signed graphs-Gallai signed graph, product-Gallai signed graph and dot-Gallai signed graph. We find the forbidden subgraph characterizations of Gallai signed graph, product-Gallai signed graph and dot-Gallai signed graph.

Let T be a tree of order n. For any edge labeling f : E → {1, 2, 3, ...} the weight of a path P is the sum of the labels of the edges of P and is denoted by w(P). If the weights of the n C 2 paths in T are exactly 1, 2,..., n C 2 , then f is called a Leech labeling and a tree which admits a Leech labeling is called a Leech tree. In this paper we determine all Leech trees having diameter three. We also prove that the tree obtained from the path P n = (v 1 , v 2 , ..., v n) by attaching a pendent vertex at v n−1 is not a Leech tree for all n ≥ 4.

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The graph G is said to be strongly regular with parameters (n, k, λ, µ) if the following conditions hold: (1) each vertex has k neighbours; (2) any two adjacent vertices of G have λ common neighbours; (3) any two non-adjacent vertices of G have µ common neighbours. In this paper we study two weaker notions of strongly regular graphs. A graph satisfying the conditions (1) and ( 2) is called an edge-regular graph with parameters (n, k, λ). We call a graph satisfying the conditions (1) and (3) a pseudo strongly regular graph with parameters