Discrete Mathematics

Graph

Vertex Discrete Mathematics Degree Graphs Isomorphism

Graph Matrices Graph as (c) Marcin Sydow Relation

Paths and Cycles

Connectedness

Trees Contents

Discrete Mathematics Introduction

Vertex Degree of a vertex Degree Graph isomorphism Isomorphism

Graph Adjacency and Incidence Matrices Matrices Graphs vs Relations Graph as Relation Path and Cycle Paths and Cycles Connectedness Connectedness Weakly and strongly connected components Trees Tree Rooted tree Binary tree Introduction

Discrete Mathematics

Graph The role of graphs:

Vertex Degree extremely important in computer science and mathematics Isomorphism numerous important applications Graph Matrices modeling the concept of binary relation Graph as Graphs are extensively and intuitively to convey information in Relation visual form. Paths and Cycles Here we introduce basic mathematical view on graphs. Connectedness

Trees Graph (the mathematical deﬁnition)

Discrete Mathematics

(c) Marcin Sydow Graph (undirected graph) is an ordered pair of sets: Graph G = (V , E), where: Vertex Degree 1 Isomorphism V is the vertex set Graph E is the edge set Matrices

Graph as each edge e = {v, w} in E is an unordered pair of Relation vertices from V , called the ends of the edge e. Paths and Cycles

Connectedness Vertex can be also called node.

Trees

1plural form: vertices Edges and vertices

Discrete Mathematics

(c) Marcin Sydow For an edge e = {v, w} ∈ E we say: Graph the edge e connects the vertices v i w Vertex Degree the vertices v and w are neighbours or are adjacent in the Isomorphism graph G Graph Matrices the edge e is incident to the vertex v (or w). Graph as Relation a self-loop is an edge of the form (v, v).

Paths and Cycles If V and E are empty G is the zero graph, if E is empty it is an Connectedness empty graph Trees Directed graph (digraph) (mathematical deﬁnition)

Discrete Mathematics

(c) Marcin Sydow Directed graph (digraph) is an ordered pair: G = (V , E), Graph where: Vertex Degree V is the vertex set Isomorphism

Graph E is the edge set (or arc set) Matrices

Graph as each edge e = (v, w) in E is an ordered pair of vertices Relation from V , called the tail and head end of the edge e, Paths and Cycles respectively. Connectedness Example Trees Simple graphs, multigraphs and hypergraphs

Discrete Mathematics

Graph Simple graph: a graph where there are no self-loops (edges or Vertex Degree arcs of the form (v, v)). Isomorphism If there are possible multiple edges or arcs between the same Graph Matrices pair of vertices we call it a multi-graph.

Graph as Relation Notice: in a directed graph (v, w) is a diﬀerent arc than (w, v)

Paths and for v 6= w. Cycles

Connectedness

Trees Picture of a graph

Discrete Mathematics

Graph A given graph can be depicted on a plane (or other Vertex Degree 2-dimensional surface) in multiple ways (example). Isomorphism A picture is only a visual form of representation of a graph. Graph Matrices It is necessary to distinguish between an abstract Graph as Relation (mathematical) concept of a graph and its picture (visual Paths and representation) Cycles

Connectedness

Trees Degree of a vertex

Discrete Mathematics

Graph Degree of a vertex v denoted as deg(v) is the number of edges Vertex Degree (or arcs) incident with this vertex. Isomorphism (note: we assume that each self-loop (v, v) contributes 2 to the Graph Matrices degree of the vertex v)

Graph as Relation If deg(v) = 0 we call it an isolated vertex. Paths and Example Cycles

Connectedness

Trees Proof: each edge contributes 2 to the sum of degrees. Corollary: sum of degrees is twice the number of edges Corollary: the number of vertices with odd degree must be even. Example

Degree sum theorem (hand-shake theorem)

Discrete Mathematics

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees Degree sum theorem (hand-shake theorem)

Discrete Mathematics

Isomorphism Proof: each edge contributes 2 to the sum of degrees.

Graph Matrices Corollary: sum of degrees is twice the number of edges

Graph as Relation Corollary: the number of vertices with odd degree must be

Paths and even. Cycles

Connectedness Example

Trees Degrees in directed graphs

Discrete Mathematics

Graph

Vertex In directed graphs: indegree of a vertex v (indeg(v)): number Degree of arcs that v is the head of Isomorphism

Graph outdegree of a vertex v (outdeg(v)): number of arcs that v is Matrices the tail of Graph as Relation Example Paths and Cycles

Connectedness

Trees Degree sum theorem for digraphs

Discrete Mathematics

Graph The sum of indegrees of all vertices is equal to the sum of Vertex Degree outdegrees of all vertices in any directed graph. Isomorphism Proof: each arc contributes 1 to the indegree sum and 1 to the Graph Matrices outdegree sum.

Graph as Relation Corollary: sum of indegrees (outdegrees) is equal to the number

Paths and of arcs in a digraph. Cycles

Connectedness

Trees Graph Isomorphism

Discrete Mathematics

Vertex v, w are connected by an edge (arc) in G1 ⇔ Degree f (v), f (w) are connected by an edge (arc) in G2. Isomorphism Graph The function f is called isomorphism between graphs G and Matrices 1

Graph as G2. Relation Example Paths and Cycles Interpretation: graphs are isomorphic if they are “the same” Connectedness from the point of view of the graph theory (they can have Trees diﬀerent names of vertices or be diﬀerently depicted, etc.). Subgraph and induced graph

Discrete Mathematics

(c) Marcin Sydow Subgraph of graph G = (V , E) is a graph H = (V 0, E 0) so that Graph V 0 ⊆ V and E 0 ⊆ E and any edge from E 0 has both its ends in Vertex 0 Degree V . Isomorphism Example Graph Matrices A subgraph of G induced by a set of vertices V 0 ⊆ V is a Graph as 0 0 Relation subgraph G of G whose vertex set is V whose edges (arcs) are 0 Paths and all edges (arcs) of G that have both ends in V . Cycles

Connectedness Example

Trees empty graph Nn (n vertices, no edges) (example)

full graph Kn (a simple graph of n vertices and all possible edges (arcs)) (example) bi-partite graph (its set of vertices can be divided into two disjoint sets so that any edges (arcs) are only between the sets) (example)

full bi-partite graph Km,n (a bipartite graph that has all possible edges (arcs))

path graph Pn (example)

cyclic graph Cn (example)

Some important graph families

Discrete Mathematics (c) Marcin (all graphs below are simple graphs) Sydow

Graph

Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees full graph Kn (a simple graph of n vertices and all possible edges (arcs)) (example) bi-partite graph (its set of vertices can be divided into two disjoint sets so that any edges (arcs) are only between the sets) (example)

full bi-partite graph Km,n (a bipartite graph that has all possible edges (arcs))

path graph Pn (example)

cyclic graph Cn (example)

Some important graph families

Discrete Mathematics (c) Marcin (all graphs below are simple graphs) Sydow empty graph Nn (n vertices, no edges) (example) Graph

Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees bi-partite graph (its set of vertices can be divided into two disjoint sets so that any edges (arcs) are only between the sets) (example)

full bi-partite graph Km,n (a bipartite graph that has all possible edges (arcs))

path graph Pn (example)

cyclic graph Cn (example)

Some important graph families

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees full bi-partite graph Km,n (a bipartite graph that has all possible edges (arcs))

path graph Pn (example)

cyclic graph Cn (example)

Some important graph families

Discrete Mathematics (c) Marcin (all graphs below are simple graphs) Sydow empty graph Nn (n vertices, no edges) (example) Graph full graph Kn (a simple graph of n vertices and all possible Vertex Degree edges (arcs)) (example) Isomorphism bi-partite graph (its set of vertices can be divided into two Graph Matrices disjoint sets so that any edges (arcs) are only between the Graph as sets) (example) Relation

Paths and Cycles

Connectedness

Trees path graph Pn (example)

cyclic graph Cn (example)

Some important graph families

Paths and full bi-partite graph Km,n (a bipartite graph that has all Cycles possible edges (arcs)) Connectedness

Trees cyclic graph Cn (example)

Some important graph families

Paths and full bi-partite graph Km,n (a bipartite graph that has all Cycles possible edges (arcs)) Connectedness path graph Pn (example) Trees Some important graph families

Paths and full bi-partite graph Km,n (a bipartite graph that has all Cycles possible edges (arcs)) Connectedness path graph Pn (example) Trees cyclic graph Cn (example) Some important graph families

Paths and full bi-partite graph Km,n (a bipartite graph that has all Cycles possible edges (arcs)) Connectedness path graph Pn (example) Trees cyclic graph Cn (example) Adjacency Matrix

Discrete Mathematics

Graph For a graph G = (V , E), having n vertices its adjacency Vertex Degree matrix is a square matrix A having n rows and columns indexed Isomorphism by the vertices so that A[i, j] = 1 ⇔ vertices i, j are adjacent, Graph Matrices else A[i, j] = 0.

Graph as (in case of self-loop (i, i), A[i, i] = 2) Relation

Paths and Example Cycles

Connectedness

Trees for undirected graphs the matrix is symmetric (AT = A) for simple graphs the diagonal of A contains only zeros sum of numbers in a row i: degree of i (outdegree for digraphs) sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Graph

Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees the matrix is symmetric (AT = A) for simple graphs the diagonal of A contains only zeros sum of numbers in a row i: degree of i (outdegree for digraphs) sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

(c) Marcin Some simple relations concerning properties of a graph and Sydow properties of its adjacency matrix: Graph for undirected graphs Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees for simple graphs the diagonal of A contains only zeros sum of numbers in a row i: degree of i (outdegree for digraphs) sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

(c) Marcin Some simple relations concerning properties of a graph and Sydow properties of its adjacency matrix: Graph for undirected graphs the matrix is symmetric (AT = A) Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees the diagonal of A contains only zeros sum of numbers in a row i: degree of i (outdegree for digraphs) sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees sum of numbers in a row i: degree of i (outdegree for digraphs) sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees degree of i (outdegree for digraphs) sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

(c) Marcin Some simple relations concerning properties of a graph and Sydow properties of its adjacency matrix: Graph for undirected graphs the matrix is symmetric (AT = A) Vertex Degree for simple graphs the diagonal of A contains only zeros Isomorphism sum of numbers in a row i: Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees sum of numbers in a column i: degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Graph as Relation

Paths and Cycles

Connectedness

Trees degree of i (indegree for digraphs) for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Paths and Cycles

Connectedness

Trees for directed graphs AT reﬂects the graph with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Paths and digraphs) Cycles

Connectedness

Trees with all the arcs “inversed”

Examples

Some Simple Observations

Discrete Mathematics

Paths and digraphs) Cycles for directed graphs AT reﬂects the graph Connectedness

Trees Examples

Some Simple Observations

Discrete Mathematics

Paths and digraphs) Cycles for directed graphs AT reﬂects the graph with all the arcs Connectedness “inversed” Trees Some Simple Observations

Discrete Mathematics

Paths and digraphs) Cycles for directed graphs AT reﬂects the graph with all the arcs Connectedness “inversed” Trees Examples Incidence matrix

Discrete Mathematics

Graph An incidence matrix I of an undirected graph G: the rows

Vertex correspond to vertices and columns correspond to edges (arcs). Degree I [v, e] = 1 ⇔ v is incident with e (else I[v,e]=0) Isomorphism

Graph Example Matrices

Graph as For directed graphs: the only diﬀerence is the distinction Relation between v being the head (=1) or the tail (=-1) of e Paths and Cycles Example Connectedness

Trees Graphs vs relations

Discrete Mathematics

Graph Each directed graph naturally represents any binary relation

Vertex R ∈ V × V . (i.e. E is the set of all pairs of elements from V Degree that are in the relation) Isomorphism

Graph Example Matrices

Graph as Each undirected graph naturally represents any symmetric Relation binary relation Paths and Cycles Example Connectedness

Trees self-loop on each vertex symmetric relation: undirected graph or always mutual arcs transitive relation: for any path there is a “short” arc anti-symmetric relation: no mutual arcs, always self-loops inverse of the relation: each arc is inversed

Observations on analogies between relations and graphs

Discrete Mathematics

Graph Vertex reﬂexive relation: Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees undirected graph or always mutual arcs transitive relation: for any path there is a “short” arc anti-symmetric relation: no mutual arcs, always self-loops inverse of the relation: each arc is inversed

Observations on analogies between relations and graphs

Discrete Mathematics

Graph Vertex reﬂexive relation: self-loop on each vertex Degree

Isomorphism symmetric relation:

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees for any path there is a “short” arc anti-symmetric relation: no mutual arcs, always self-loops inverse of the relation: each arc is inversed

Observations on analogies between relations and graphs

Discrete Mathematics

Graph Vertex reﬂexive relation: self-loop on each vertex Degree

Isomorphism symmetric relation: undirected graph or always mutual arcs Graph transitive relation: Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees no mutual arcs, always self-loops inverse of the relation: each arc is inversed

Observations on analogies between relations and graphs

Discrete Mathematics

Graph Vertex reﬂexive relation: self-loop on each vertex Degree

Isomorphism symmetric relation: undirected graph or always mutual arcs Graph transitive relation: for any path there is a “short” arc Matrices

Graph as anti-symmetric relation: Relation

Paths and Cycles

Connectedness

Trees each arc is inversed

Observations on analogies between relations and graphs

Discrete Mathematics

Graph Vertex reﬂexive relation: self-loop on each vertex Degree

Isomorphism symmetric relation: undirected graph or always mutual arcs Graph transitive relation: for any path there is a “short” arc Matrices

Graph as anti-symmetric relation: no mutual arcs, always self-loops Relation inverse of the relation: Paths and Cycles

Connectedness

Trees Observations on analogies between relations and graphs

Discrete Mathematics

Graph Vertex reﬂexive relation: self-loop on each vertex Degree

Isomorphism symmetric relation: undirected graph or always mutual arcs Graph transitive relation: for any path there is a “short” arc Matrices

Graph as anti-symmetric relation: no mutual arcs, always self-loops Relation inverse of the relation: each arc is inversed Paths and Cycles

Connectedness

Trees Path

Discrete Mathematics

(c) Marcin Sydow Path: an alternating sequence of vertices and edges Graph (v0, e0, v1, eq,..., vk , ek ,..., vl ) so that each edge ek is incident Vertex Degree with vertices vk , vk+1. We call it a path from v0 to vl . Isomorphism (sometimes it is convenient to deﬁne path just as a subsequence Graph Matrices of vertices or edges of the above sequence) Graph as Relation Example

Paths and Cycles Directed path in a directed graph is deﬁned analogously (the Connectedness arcs must be directed from vk to vk+1 Trees Paths cont.

Discrete Mathematics

Graph simple path: no repeated edges (arcs) Vertex Degree elementary path: no repeated vertices Isomorphism Examples Graph Matrices length of a path: number of its edges (arcs) Graph as Relation (assume: 0-length path is a single vertex) Paths and Cycles Example Connectedness

Trees Distance in graph

Discrete Mathematics

Graph path between them.

Vertex Degree The distance function in graphs d : V × V → N has the

Isomorphism following properties:

Graph Matrices d(u, v) = 0 ⇔ u == v Graph as (only in undirected graphs) it is a symmetric function, i.e. Relation ∀u, v ∈ V d(u,v) = d(v,u) Paths and Cycles triangle inequality: ∀u, v, w ∈ V it holds that Connectedness d(u, v) + d(v, w) ≥ d(u, w) Trees Cycle

Discrete Mathematics

Graph cycle: a path of length at least 3 (2 for directed graphs) where Vertex the beginning vertex equals the ending vertex v0 == vl (also Degree called a closed path) Isomorphism

Graph Example Matrices

Graph as analogously: directed cycle, simple cycle, elementary cycle Relation (except the starting and ending vertices there are no repeats) Paths and Cycles Examples Connectedness

Trees Connectedness

Discrete Mathematics

Graph

Vertex Degree A graph is connected ⇔ for any two its vertices v,w there Isomorphism exists a path from v to w Graph Matrices Example Graph as Relation

Paths and Cycles

Connectedness

Trees Connected component of a graph

Discrete Mathematics

Graph

Vertex Degree Connected component of a graph is its maximal subgraph Isomorphism that is connected. Graph Matrices Example (why “maximal”)? Graph as Relation

Paths and Cycles

Connectedness

Trees Strongly connected graph

Discrete Mathematics

Graph A directed graph is stronlgy connected ⇔ for any pair of its

Vertex vertices v,w there exists a directed path from v to w. Degree

Isomorphism Example Graph A directed graph is weakly connected ⇔ for any pair of its Matrices

Graph as vertices v,w there exists undirected path from v,w (i.e. the Relation directions of arcs can be ignored) Paths and Cycles note: strong connectedness implies weak connectedness (but Connectedness not the opposite) Trees Example Strongly and weakly connected components

Discrete Mathematics

Graph

Vertex Strongly connected component: a maximal subgraph that is Degree strongly connected Isomorphism

Graph Weakly connected component: a maximal subgraph that is Matrices weakly connected Graph as Relation Examples Paths and Cycles

Connectedness

Trees Drzewa

Discrete Mathematics

(c) Marcin Sydow Tree is a graph that is connected and does not contain cycles (acyclic). Graph Vertex Example Degree Isomorphism Forest is a graph that does not contain cycles (but does not Graph have to be connected) Matrices Graph as Example Relation

Paths and A leaf of a tree is a vertex that has degree 1. Cycles

Connectedness Other vertices (nodes) are called internal nodes of a tree. Trees Example T is a tree of n vertices T has exactly n-1 edges (arcs) and is acyclic T is connected and has exactly n-1 edges (arcs) T is connected and removing any edge (arc) makes it not connected any two vertices in T are connected by exactly one elementary path T is acyclic and adding any edge makes exactly one cycle

Equivalent deﬁnitions of a tree

Discrete Mathematics

Graph

Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees T has exactly n-1 edges (arcs) and is acyclic T is connected and has exactly n-1 edges (arcs) T is connected and removing any edge (arc) makes it not connected any two vertices in T are connected by exactly one elementary path T is acyclic and adding any edge makes exactly one cycle

Equivalent deﬁnitions of a tree

Discrete Mathematics

Graph T is a tree of n vertices Vertex Degree

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees T is connected and has exactly n-1 edges (arcs) T is connected and removing any edge (arc) makes it not connected any two vertices in T are connected by exactly one elementary path T is acyclic and adding any edge makes exactly one cycle

Equivalent deﬁnitions of a tree

Discrete Mathematics

Graph T is a tree of n vertices Vertex Degree T has exactly n-1 edges (arcs) and is acyclic

Isomorphism

Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees T is connected and removing any edge (arc) makes it not connected any two vertices in T are connected by exactly one elementary path T is acyclic and adding any edge makes exactly one cycle

Equivalent deﬁnitions of a tree

Discrete Mathematics

Graph T is a tree of n vertices Vertex Degree T has exactly n-1 edges (arcs) and is acyclic Isomorphism T is connected and has exactly n-1 edges (arcs) Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees any two vertices in T are connected by exactly one elementary path T is acyclic and adding any edge makes exactly one cycle

Equivalent deﬁnitions of a tree

Discrete Mathematics

Graph T is a tree of n vertices Vertex Degree T has exactly n-1 edges (arcs) and is acyclic Isomorphism T is connected and has exactly n-1 edges (arcs) Graph Matrices T is connected and removing any edge (arc) makes it not Graph as connected Relation

Paths and Cycles

Connectedness

Trees T is acyclic and adding any edge makes exactly one cycle

Equivalent deﬁnitions of a tree

Discrete Mathematics

Paths and any two vertices in T are connected by exactly one Cycles elementary path Connectedness

Trees Equivalent deﬁnitions of a tree

Discrete Mathematics

Paths and any two vertices in T are connected by exactly one Cycles elementary path Connectedness T is acyclic and adding any edge makes exactly one cycle Trees Rooted tree

Discrete Mathematics

Vertex Example Degree

Isomorphism Distinguishing the root introduces a natural hierarchy among

Graph the nodes of the tree: the lower the depth the higher the node Matrices in the hierarchy. Graph as Relation Picture of a rooted tree: root is at the top, all nodes of the Paths and Cycles same depth are on the same level, the higher the depth, the Connectedness lower the level on the picture. Trees Example Terminology of rooted trees

Discrete Mathematics

(c) Marcin A depth of a vertex v of a rooted tree, denoted as depth(v) is Sydow its distance from the root. Graph Height of a rooted tree: maximum depth of any its node Vertex Degree ancestor of a vertex v is any vertex w that lies on any path Isomorphism from the root to v, v is then called a descendant of w (the Graph Matrices root does not have ancestors and the leaves do not have

Graph as descendants) Relation

Paths and a ancestor w of a neighbour (adjacent) vertex v is called the Cycles parent of v, in this case v is called the child of w. Connectedness

Trees if vertices u, v have a common parent we call them siblings Examples Binary tree

Discrete Mathematics

Graph Binary tree is a rooted tree with the following properties: Vertex Degree

Isomorphism each node has maximally 2 children Graph for each child it is speciﬁed whether it is left or right child Matrices of its parent (max. 1 left child and 1 right child) Graph as Relation Paths and Example Cycles

Connectedness

Trees Summary

Discrete Mathematics

(c) Marcin Sydow Mathematical deﬁnition of Graph and Digraph Graph Degree of a vertex Vertex Degree Graph isomorphism Isomorphism Adjacency and Incidence Matrices Graph Matrices Graphs vs Relations

Graph as Relation Path and Cycle Paths and Connectedness Cycles

Connectedness Weakly and strongly connected components Trees Tree, Rooted tree, Binary tree Example tasks/questions/problems

Discrete Mathematics give the mathematical deﬁnitions and basic properties of the (c) Marcin discussed concepts and their basic properties (in particular: Sydow graph, digraph, degree, isomorphism, adjacency/incidence

Graph matrix, path and cycle, connectedness and connected components, trees (including rooted and binary trees) Vertex Degree make picture of the speciﬁed graph of one of the discussed Isomorphism families (full, bi-partite, etc.) Graph Matrices given a picture of a graph provide its mathematical form (pair Graph as of sets) and adjacency/incidence matrix and vice versa Relation Paths and check whether the given graphs are isomorphic and prove your Cycles answer Connectedness

Trees ﬁnd connected components of a given graph (or weakly/strongly connected components for a digraph) specify the height, depth, number of leaves, etc. of a given rooted tree Discrete Mathematics

Graph

Vertex Degree Isomorphism Thank you for your attention. Graph Matrices

Graph as Relation

Paths and Cycles

Connectedness

Trees